Spectral Gap Estimates on Conformally Flat Manifolds
Gabriel Khan Malik Tuerkoen

A Hard Energy Spectrum in 3D Guide-Field Magnetic Reconnection
    Masahiro Hoshino
    April 29, 2024
==============================================================


The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schrödinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic space, Nguyen, Stancu and Wei conjectured that it is possible to obtain a lower bound on the fundamental gap in terms of the diameter of the domain and the dimension <cit.>. In this article, we prove this conjecture by establishing conformal log-concavity estimates for the first eigenfunction. This builds off earlier work by the authors and Saha <cit.> as well as recent work by Cho, Wei and Yang <cit.>. We also prove spectral gap estimates for a more general class of problems on conformally flat manifolds and investigate the relationship between the gap and the inradius. For example, we establish gap estimates for domains in 𝕊^1 ×𝕊^N-1 which are convex with respect to the universal affine cover.





§ INTRODUCTION
 



The goal of this paper is to establish lower bounds on the fundamental gap of sufficiently convex domains in conformally flat manifolds, with a particular focus on horoconvex domains in hyperbolic space. Given a bounded domain Ω in a Riemannian manifold (M,g), we consider the Dirichlet eigenvalue problem

    -Δ _g u = λ(Ω) u   Ω,      u = 0   ∂Ω,

where Δ_g is the Laplace operator.
There is a discrete sequences of eigenvalues 

    0<λ_1(Ω) ≤λ_2(Ω)≤⋯≤λ_k(Ω)≤⋯→ +∞,

repeated with multiplicity.
The fundamental gap is the difference between the first and second eigenvalues

    Γ(Ω) = λ_2(Ω)-λ_1(Ω)>0,

which is non-zero whenever the domain is connected. For convex domains in ℝ^N,
Andrews and Clutterbuck <cit.> proved that the fundamental gap of the Schrödinger operator -Δ + V is greater than the Dirichlet gap of a one-dimensional model: 

    Γ(Ω, V) ≥λ_2(V) - λ_1(V).

Here, λ_i is the i-th eigenvalue of the operator ℒ = -d^2ds^2+V on [-D/2,D/2], and V' is a modulus of convexity for V. In other words, for any x,y∈Ω, we have that 

    ⟨∇ V(y), γ_x,y'(d(x,y)2)⟩ -  ⟨∇ V(x), γ_x,y'(-d(x,y)2)⟩≥ 2 V'(d(x,y)2),

 where γ_x,y is the minimizing geodesic such that γ_x,y(-d(x,y)2) = x and γ_x,y(d(x,y)2) = y.

There has been a large amount of work studying the fundamental gap in other geometries as well.
For convex regions on the sphere 𝕊^N, Dai, He, Seto, Wang and Wei (in various subsets) proved the analogous result to the Andrews-Clutterbuck theorem
<cit.> (see also <cit.> for earlier work). Recently the authors along with Nguyen, Saha and Wei (in various subsets) made progress in obtaining spectral gap estimates for deformations of round spheres in dimension two <cit.> and for conformal deformations of the sphere in all dimensions <cit.>.

 For spaces of negative curvature, the behavior of the fundamental gap of convex domains can be remarkably different compared to flat or spherical geometry. In fact, Bourni et al. <cit.> showed that for any D>0 there are convex domains Ω of any diameter D in hyperbolic N-space ℍ^N   whose spectral gap is arbitrarily small. The first-named author and Nguyen then extended this result to spaces of mixed sectional curvature (for small diameter) and negatively pinched curvature for arbitrary diameter <cit.>.

Since assuming convexity is not enough to ensure uniform lower bounds on the fundamental gap in such geometries, it is natural to consider domains which satisfy stronger convexity assumptions. In particular, Nguyen, Stancu and Wei  <cit.> conjectured that there is a lower bound on the fundamental gap of domains which are horoconvex (see Definition <ref> for the definition of horoconvexity).


    There exists a constant c(N,D)>0, such that for any horoconvex domain Ω⊂ℍ^N, the following inequality holds:
    
    Γ(Ω) ≥ c(N,D).


In recent work <cit.>, the authors along with Saha showed that for N  = 2 and D ≤ 2  (2 √(11/3) ), the gap satisfies Γ(Ω) ≥323 (7 + √(33))π^2 D^2 + 43. Therefore, the conjecture holds in two-dimensions when the diameter is not too large.
    
In this article, we drop the restrictions on the the dimension and the diameter in order to prove the general conjecture. 
 
    Conjecture <ref> holds.
 

Apart from this question, we are able to obtain fundamental gap estimates for domains in conformally flat manifolds which are convex with respect to a flat connection. 

    Suppose that Ω⊂ (M, g = e^2φg_) is a convex domain with respect to Euclidean geometry, where g_ denotes the Euclidean metric. Then the fundamental gap of Ω satisfies 
    Γ(Ω, Δ_g) ≥minρ/max_Ωexp(2φ)Γ(ρ, V)-N-2/4(N-1)(max_Ω R_g -min_Ω R_g),

    where R_g denotes the scalar curvature and ρ and V which can be solved explicitly in terms of φ and λ_1(Ω).
 


It is worthwhile to compare this result to those Oden-Song-Wang and Ramos et al. <cit.>, which establish lower bounds on the fundamental gap for domains which satisfy an interior rolling ball condition. However, this is a comparatively restrictive class of domains and requires that the boundary is uniformly C^2. On the other hand, our estimates require conformal flatness but only depend on the diameter, conformal factor (and possibly the inradius).



The strategy to prove these results is to study the eigenfunctions of the problem (<ref>) with respect to Euclidean metric, using a conformal deformation. More generally, if g̃ = e^2φg and ũ is a g̃ Dirichlet eigenfunction, then u = e^N-22φũ solves a Schrödinger equation of the form  
    -Δ_g u + V u = λ(Ω, ρ, V) ρ u   Ω,      u = 0   ∂Ω,

where V is a potential and ρ is a weighting function (see Section <ref> for details). Therefore, we can establish estimates on the gap of domains in conformally flat geometry by studying them in terms of an underlying conformal connection (which will typically be chosen to have constant or vanishing curvature).


To obtain spectral gap estimates, we compare the gap of the problem of (<ref>) to the one-dimensional problem:  

    -φ”+ Vφ= λ(ρ, V)ρφ  [- D2,  D2],     φ(- D2) = φ(  D2) = 0

where V' is a modulus of convexity for V and ρ' is a modulus of concavity of ρ. 
 More precisely, the main theorem which provides the gap estimates is the following.

    Let Ω⊂ (M,g_𝕄^N_K) be a convex domain, where g_𝕄^N_K denotes the metric of the space of constant sectional curvature K ≥ 0. Suppose that ρ' is a modulus of concavity for ρ and that V is a modulus of convexity of V. Moreover, assume the additional inequalities
    
    [ρ(s) _K^2(s)]' (λ(V, ρ)-λ_1(Ω, ρ, V))≥ 0,

     where _K(s) = cos(√(K) s) and (if K>0) that
    
    ρ≤minρ,   V≥max V.

    Then the spectral gap of the problem (<ref>) satisfies
    
    Γ(Ω, ρ, V) ≥minρ/max_ΩρΓ(ρ, V),

    where Γ(ρ, V) is the spectral gap of (<ref>).


This result generalizes previous results from <cit.>, which considered the case ρ≡ 1. A major step of the proof is to establish a modulus of concavity for the log of the principle eigenfunction of (<ref>). In essence, this estimate shows that the principle Dirichlet eigenfunction is at least as log-concave as the solution to the one-dimensional model.  
 
    Under the assumptions of Theorem <ref>, we have that for v = log u_1 that for  x,y ∈Ω
    
    ⟨∇ v(y), γ_x,y'(d(x,y)2)⟩ -⟨∇ v(x), γ_x,y'(-d(x,y)2)⟩≤ 2(logφ_1)'( d(x,y)2)+(N-1)_K(d(x,y)2)

   where φ_1 is the first eigenfunction of the problem <ref> and _K is defined as _K(s)=√(K)tan (√(K)s).

It is well-known that log-concavity estimates for the principle eigenfunction imply lower bounds on the fundamental gap (see, e.g., <cit.>). 
Dividing (<ref>) by d(x,y) and passing to the limit as d(x,y) → 0^+, 
gives log u_1 ≤ (logφ_1)”(0)+(N-1)K2. Using this upper bound one can derive a lower bound of Γ(Ω) using known Neumann eigenvalue comparisons <cit.> (see Section <ref>).
In the literature, there are currently two methods to prove (<ref>). One can either use a maximum principle argument (see e.g. <cit.>) or a stochastic analysis argument using coupled diffusions. In this paper, we use the stochastic method, but with a slight variation it is possible to obtain the same result using the maximum principle (see Subsubsection <ref> for a brief discussion of this). The diffusion approach was introduced in <cit.>, where they gave an alternative proof of the fundamental gap conjecture. Recently, Cho, Wei and Yang <cit.> generalized this technique to spherical geometry, and our work follows their approach closely.






The assumptions (<ref>) and (<ref>) in Theorem <ref> appear to be technical. However, Condition (<ref>) is necessary in the following sense: in hyperbolic space, there are domains which are convex with respect to the flat connection with very small inradius (i.e., large eigenvalue) but whose spectral gap is arbitrarily small (see Section <ref> and the Appendix <ref> for details). Therefore, it is necessary for our estimates to decay as the eigenvalue becomes large. In fact, we show that if the inradius is bounded below, the spectral gap is bounded from below by a positive constant (see Corollary <ref>).

In the case that ρ is concave, 

we can take ρ to be a constant (or even decreasing), in which case (<ref>) is not a major obstacle and the functions ρ and V can be chosen to only depend on the diameter D and the conformal factor exp(2φ) (see e.g. (<ref>)). When the underlying geometry is hyperbolic space (or 𝕊^1 ×𝕊^N-1), the weighting function ρ will not be concave, which will affect the estimates that can be established and force V (and thus the gap estimate) to depend on the first eigenvalue λ_1(Ω). 






 §.§ Overview of the Paper



In Section <ref>, we briefly review some background on hyperbolic and conformal geometry that will be used throughout the paper. In particular, we show that one can bound fundamental gaps a conformally flat metric in terms of the spectral gap of a related problem in Euclidean metric (see Proposition <ref>). In Subsection <ref>, we consider the two-point function Z(x,y) defined by (with respect to the Levi-Civita connection of g_𝕄^N_K)

    Z(x,y) = ⟨∇ v(y) , γ_x,y'(d(x,y)2)⟩ - ⟨∇ v(x) , γ_x,y'(-d(x,y)2)⟩,

where γ_x,y denotes the geodesic such that γ_x,y(-d(x,y)2) = x and γ_x,y(d(x,y)2) = y and compute its derivatives for v = log u_1, where u_1 denotes the first eigenfunction of the problem (<ref>). The computation is similar to the one presented in <cit.> but here we calculate this at any point (x,y), not necessarily a maximal point (see also <cit.>).

In Section <ref>, we first review some basic facts about stochastic analysis on manifolds. We then prove Theorem <ref> via a coupled diffusion method, following <cit.>. 
  In Section <ref>, we study the eigenvalues of the model (<ref>) and prove a lower bound on the fundamental gap in dimension one, which we will then use for the comparison of the gap. In Section <ref>, we turn our attention to horoconvex domains and prove Theorem <ref>. We then derive a more explicit lower bound of the spectral gap for horoconvex domains using a comparison of Neumann eigenvalues. More precisely, we consider w= u_2/u_1, which satisfies

    Δ w +2∇log u_1·∇ w=-Γ(Ω, ρ)ρ w

    with Neumann boundary conditions <cit.>. Then from Proposition  <ref> one obtains an upper bound on the Hessian of v = log u which then induces a lower bound on the Bakry-Émery-Ricci curvature <cit.>.
    
 In Section <ref>, we prove Theorem <ref>. We also investigate the relationship between the inner radius of a given domain Ω and its spectral gap. In particular, we show that if the conformal factor is concave with respect to the Euclidean metric g_ , one can prove lower bounds for g_-convex domains that are independent of the inner radius. However, in general it is only possible to prove lower bounds depending on the diameter if we restrict ourselves to domains which are g_ convex and whose inradius is controlled by the diameter (see Section <ref> for more details). We then prove a lower bound on the fundamental gap for small conformal deformation of the round sphere (see Theorem <ref>).




 §.§ Acknowledgements


Malik T. wishes to thank Guofang Wei for her helpful comments and guidance.  G. Khan would like to thank David Herzog for some helpful conversations. He is supported in part by Simons Collaboration Grant 849022.




§ BACKGROUND AND PRELIMINARIES





 §.§ Horospheres and horoconvexity


Since Theorem <ref> concerns horoconvex domains in hyperbolic geometry, we review some basic properties of such domains.

In hyperbolic space, a horosphere is a specific type of hypersurface which can be formed as the limit of expanding balls in hyperbolic space which share a tangent hyperplane and its point of tangency. Equivalently, one can consider them as spheres whose center lies on the ideal boundary of ℍ^N. In the Poincaré disk model of hyperbolic space, horospheres are Euclidean spheres inside the unit ball B which are tangent to ∂ B.




We say that a domain Ω⊂ℍ^N is horoconvex if at every point p ∈∂Ω there exists a horosphere passing through p which contains Ω entirely.


Equivalently, a domain Ω is horoconvex if and only if all the principal
curvatures of its boundary hypersurface are at least one. In this way, horoconvexity is a strengthening of convexity.





 §.§ Conformal deformations of the Laplace operator


We now review some properties of conformal geometry. Given a function φ: M→ℝ, we consider the conformal metric  g̃ = e^2φg. There are known formulas the Hessian and the Laplace operator of under conformal change, which states that for a smooth function F:M →ℝ,

    _g̃ F    = _g F - 2 dφ⊗ d F +(∇φ·∇ F) g 
    Δ _g̃F    = e^-2φ(Δ_g F +(N-2)∇φ·∇ F).

 Moreover, the scalar curvature R_g of the conformal metric satisfies: 

    R_e^2φg  = e^-2φ(R_g-2(N-1) Δ_g φ-(N-1)(N-2)|∇φ|^2).



Now, we consider a  g̃-eigenfunction ψ satisfying

    Δ_g̃ψ = -λψ  Ω    ψ = 0   ∂Ω.

A straightforward argument shows the function u = ψ e^N-22φ is a weighted eigenfunction of the following Schrödinger operator of g.

    -Δ_g u +[ (N-2)^24  |∇φ |^2+N-22Δ_g φ]u = λ   e^2φ  u
    
          u |_∂Ω = 0
       ,

which is a weighted Schrödinger equation with weight ρ =e^2φ and potential 
    V =  (N-2)^24  |∇φ |^2+N-22Δ_g φ= - N-2/4(N-1)e^2φR_g̃+N-2/4(N-1)R_g.


In this paper, we study equations of the type (<ref>) and so we choose the conformal factor to facilitate this analysis. We denote the eigenvalues by λ_i(Ω,Δ_g, ρ, V). In case V ≡ 0, we just write λ_ i(Ω, Δ_g, ρ). If clear from the context, we omit the dependence of Δ_g. Moreover, for any function f :Ω→ℝ, we denote _Ω (f) = sup_Ω f -inf_Ω f to be its oscillation.

Let Ω⊂(M,g= e^2φg_). Then 

    Γ(Ω, Δ_g)  ≥Γ(Ω, Δ_g_, e^2φ)-N-2/4(N-1)_Ω (R_g),
 
where R_g denotes the scalar curvature of g.


    In view of (<ref>), we only need to study the eigenvalues of the problem  
    -Δũ +Vũ = λ e^2φũ  Ω,  ũ = 0   ∂Ω.

Note that 

    V = -N-2/4(N-1)e^2φR_g,

where R_g is the scalar curvature of the metric g. 
Note that for ρ = exp (2φ), the eigenvalues satisfy 

    λ_1(Ω, g)    = inf∫_Ω |∇ f|^2+Vf^2  dx/∫ _Ωρ f^2  dx
       = inf∫_Ω |∇ f|^2-N-24(N-1)ρ R_g f^2  dx/∫ _Ωρ f^2  dx.

Thus, 

    λ_i(Ω, ρ)-N-2/4(N-1)max_Ω R_g≤λ_i(Ω, g)≤λ_i(Ω, ρ) -N-2/4(N-1)inf_Ω R_g.

In particular, we have that for the fundamental gap, 

    Γ(Ω, g) ≥Γ(Ω,Δ_g_, ρ)-N-2/4(N-1)_Ω (R_g).



    The proof above shows that if (M,g) has constant scalar curvature, then we have that 
    
    λ_i(Ω, Δ_g) = λ_i(Ω, Δ_
            g_, e^2φ)-(n-2)R_g/4(n-1)




From Proposition <ref>, in the case of constant scalar curvature, it is possible to bound the fundamental gap if we can obtain a bound for the problem

    -Δ_g_ u = e^2φλ u   Ω   u = 0   ∂Ω.





 §.§ Derivatives of Two-Point Functions in Curved Geometries

In this section, we compute the second-derivatives of a two-point function Z in a Riemannian manifold. In previous works <cit.>, these derivatives were computed explicitly at the points which maximize the function. In this paper, we calculate the derivative without assuming that we are at a maximum, since this will be needed for the probabilistic argument to obtain a modulus of concavity.

Let Ω⊂ (M, g_𝕄^N_K) be uniformly convex. In other words, for any two different points x,y∈Ω, there is a unique minimizing geodesic which is contained entirely in Ω. We denote this geodesic by γ_x,y and use the convention that γ_x,y(-d(x,y)2) = x and γ_x,y(d(x,y)2) = y. We then let v = log u_1, where u_1 is the first Dirichlet eigenfuntion of the problem (<ref>) and consider the function Z:Ω×Ω→ℝ defined by (<ref>)





 Now, we let x,y be a distinct pair of points in Ω and denote the set of such points as Conf_2(Ω) (short for configuration space).

  In order to compute derivatives of Z at (x,y), we first define a frame along γ_x,y, which will be helpful in the computations. More precisely, we let {e_i}_1≤ i≤ N
 be an orthonormal basis at x with e_n=γ'_x,y(-d2) and parallel transport {e_i}_1≤ i≤ N along γ_x,y. We then define the vectors

    E_i= e_i⊕ e_i, 1≤ i≤ N-1,     E_N=e_N⊕( -e_N).
 
For notational purposes, we introduce the following notation. For a vector field X on Ω, we define the two-point function ℱ_X(x,y) as follows 

    ℱ_X(x,y) = ⟨ X(y), γ_x,y'( d2)⟩ - ⟨ X(x), γ_x,y'(-d2)⟩ .

Recall that ρ is said to be a modulus of concavity of ρ if 

    ℱ_∇ρ(x,y) ≤ 2 ρ'( d2)

and we call V a modulus of convexity of V if -V is a modulus of concavity of -V, i.e. if 

    ℱ_∇ V(x,y) ≥ 2 V'( d2).

Finally, given a function f on Ω, we define the two point function 𝒞_f(x,y) as follows 
    𝒞_f(x,y) = f(x) + f(y).



 Using the above notation, one has that 
 
    ∑_i =1 ^N ∇ ^2_E_i,E_iZ(x,y)    = -2 _K( d2)∇ _E_NZ(x,y) -2∇_∇ v(x) ⊕∇ v(y)Z(x,y)+(N-1)[K-_K^2( d2)] Z(x,y) 
         -2_K( d2)𝒞_[V-λ_1ρ](x,y)+ℱ_∇ [V-λ_1 ρ](x,y)    
          +2_K( d2)[v_N^2(x)+v_N^2(y)]+2/_K(d)∑_i =1 ^N-1(v_i(y)-v_i(x))^2.


 To compute these derivatives, we first compute ∇^2_E_N,E_NZ(x,y). Since ∇_e_Nγ'_x,y(s) =0 we immediately have


    ∇^2 _E_N,E_NZ(x,y)=⟨∇ _e_N∇ _e_N∇ v(y),e_N⟩ -⟨∇ _e_N∇ _e_N∇ v(x),e_N⟩.

 To compute the derivatives ∇ ^2_E_i,E_iZ(x,y)
 with i=1,…, N-1, we introduce variations of γ_x,y(s), which are denoted
 
    η_i(r,s): (-δ, δ) × [-d2, d2] →Ω.
 
To define this variation, we let σ_1(r) be the geodesic with σ_1(0) =x, ∂∂ rσ_1(0)=e_i(-d2) and  σ_2(r) be the geodesic with σ_2(0) =y, ∂∂ rσ_2(0)=e_i(d2).
We define η_i(r, s), s ∈ [-d2,d2] to be the minimal geodesic connecting σ_1(r) and σ_2(r).  Since Ω is strongly convex, the variation η(r,s) is smooth.
 
 
For fixed r ≠ 0, the curves η_i(r,·) will not be unit speed geodesics in general. Hence we define
   
    T_i(r,s):=η_i'/η_i',

where we denoted ∂/∂ s by ', which is a convention we will use throughout the rest of the paper.

Doing so, we have the following identity:  

    ∇^2_E_i,E_iZ(x,y)  =d^2/dr^2Z(η_i(r,-d2),η_i(r,d2))|_r=0
       =⟨∇_e_i∇ _e_i∇ v(y),e_N⟩ -⟨∇_e_i∇ _e_i∇ v(x),e_N⟩
          +2⟨∇_e_i∇ v (y),∇ _r T_i(r,d2)⟩ -2⟨∇_e_i∇ v(x_0),∇ _r T_i(r,-d2)⟩
         +⟨∇ v(y),∇ _r∇ _rT_i(0,d2)⟩ - ⟨∇ v(x),∇ _r∇ _r T_i(0,-d2)⟩.

We then denote the variation field 

    J_i(r,s) =  ∂∂ rη_i(r,s),
 which is the Jacobi field along η(s) satisfying J_i(r, -d2)  =σ_1'(r),   J_i(r,d2)  = σ_2'(r). For simplicity, we will often drop the initial 0 and denote J_i(0,s)= J_i(s). 

We can simplify this expression using several formulas derived on <cit.>. In particular, they showed that for γ = γ_x,y

    ∇ _r T_i(0,s)   =   -⟨γ',J_i'⟩ e_N+J_i', 
    ∇ _r∇ _r T_i(0,s)   =   (3⟨γ',J_i'⟩^2 -J_i'^2-⟨∇ _r∇ _r ∂η_i∂ s,e_N⟩)e_N -2⟨γ',J_i'⟩ J_i'+∇ _r∇ _r ∂η_i∂ s.

As the tangential component of a Jacobi field is linear and ⟨γ',J_i⟩=0, at the end points, we have ⟨γ',J_i⟩=0. Therefore ⟨γ',J_i'⟩=0.
Moreover, since the metric has constant sectional curvature K, we have that the Jacobi fields are given explicitly by 

    J_i(s)  = _K(s)/_K( d2)e_i(s),

so that we find 

    ∇ _r T_i(0,s)= -K_K(s)/_K( d2),    ∇ _r∇ _r T_i(0,s)=-K^2_K^2(s)/_K^2( d2)e_N+∑_j = 1^N-1⟨∇ _r∇ _r ∂η_i∂ s, e_j⟩ .

In <cit.>, it was shown that ⟨∇ _r∇ _r ∂η_i∂ s, e_j⟩ = 0 for all j = 1, …, N-1, 







(<ref>) and (<ref>), we have 




    ∇^2_E_i,E_iZ(x,y) =
    
        
        
        
           ⟨∇_e_i∇ _e_i∇ v(y),e_N⟩ -⟨∇_e_i∇ _e_i∇ v(x),e_N⟩ -2_K( d2) [v_ii(x)+v_ii(y)]    
       -_K^2( d2) Z(x,y).

We now sum these up for all i = 1, …, N and obtain that 

    ∑_i=1^N ∇^2_E_i,E_iZ(x,y) 
           =⟨Δ∇ v(y),e_N⟩ -⟨Δ∇ v(x),e_N⟩ -2_K( d2) [Δ v(y) +Δ v(x)]
          -2_K( d2)∇_E_NZ(x,y)-(N-1)_K^2( d2) Z(x,y).

Then using the identities 
    -λρ +V-Δ v    =    ∇ v^2 
    Δ∇ v -(∇ v, ·)    =    ∇Δ v,

 we find that 
 
    ∑_i=1^N ∇^2_E_i,E_iZ(x,y) 
           = -2_K( d2)𝒞_[V-λ_1ρ](x,y)+ℱ_∇[ V-λ_1ρ](x,y)    
           +2_K( d2) 𝒞_∇ v^2(x,y)-ℱ_∇∇ v^2(x,y)
          -2_K( d2)∇_E_NZ(x,y)-(N-1)_K^2( d2) Z(x,y).

We now compute -ℱ_∇∇ v^2(x,y) to show that   
    -ℱ_∇∇ v^2(x,y) = -2∇_∇ v(x) ⊕∇ v(y)Z(x,y) -2_K( d2)∑_i = 1^N-1[v_i^2(x)+v_i^2(y)]+2/_K(d)∑_i =1 ^N-1(v_i(y)-v_i(x))^2.

We choose the variation η(r,s) for the vector ∇ v(x) ⊕∇ v(y), and the unit variation field  T as before. 
We then have that 

    ∇_∇ v(x) ⊕∇ v(y)Z(x,y)    = d/drZ(η_i(r,-d2),η_i(r,d2))|_r=0
       =⟨∇ _∇ v(y)∇ v(y),e_N⟩ -⟨∇ _∇ v(x)∇ v(x),e_N⟩
          +2⟨∇ v (y),∇ _r T(r,d2)⟩ -2⟨∇ v(x_0),∇ _r T(r,-d2)⟩|_r=0.

We then let J(s) be the Jacobi field along γ with J(- d2) = ∇ v (x),  J( d2) =∇ v(y), that is, 
  
    J(s) = (  [12-sd]v_N(x)+[12+sd]v_N(y)) e_N   + ∑_i=1^N-1[v_i(x) _K( d2 -s)/_K(d)+v_i(y) _K( d2 +s)/_K(d)]e_i.

  Using (<ref>), we find that
  
    ∇_∂∂ r T(r, s)|_r=0 = ∑_i=1^N-1[-v_i(x) _K( d2 -s)/_K(d)+v_i(y)_K( d2 +s)/_K(d)]e_i.

Plugging these in, together with the fact that ∇_∇ v∇ v=1/2∇‖∇ v ‖^2, we get

    2∇ _∇ v(x) ⊕∇ v(y) Z(x,y) 
       =⟨∇∇ v(y)^2 , γ'(d2) ⟩-⟨∇∇ v(x)^2 , γ'(-d2) ⟩
          +2∑_i=1^N-1v_i(y)(_K(d)/_K(d) v_i(y) -1/_K(d)v_i(x)  )
          -2∑_i=1^N-1v_i(x)(1/_K(d)v_i(y) -_K(d)/_K(d)v_i(x)  ).

The claim then follows since by straight-forward computation, using _K( d2) = 1-_K(d)_K(d).





§ PROOF OF THEOREM <REF> VIA STOCHASTIC ANALYSIS



In order to prove Theorem <ref>, we use the technique of coupled diffusions. This method was previously used in <cit.> to prove the fundamental gap theorem in Euclidean and spherical geometry. The proof is divided into two steps. We first prove Proposition <ref> and then Proposition <ref>, a spectral gap comparison (see also <cit.>). The proof of Theorem <ref> then follows from combining these two propositions. 


 §.§ Stochastic Analysis

In order to apply coupled diffusions in this setting, we set some notation. We let O(M) denote the orthonormal frame bundle over M and π : O(M) → M be the canonical projection. Similar to <cit.>, we define H_i(U) to be the horizontal vector field on O(M), which satisfies  π_*(H_i) = U e_i, where {e_i}_i=1^N is the standard basis on ℝ^N and U ∈ O(M). 

  Following the strategy used in earlier work, we now construct a diffusion (X_t, Y_t) on the product manifold M× M using the Eells-Elworthy-Malliavin approach. 
  We first construct a diffusion (U_t,V_t) on O(M)× O(M) and then project it to M× M via π. To introduce the coupling, let m_x,y:T_xM → T_yM be the mirror map, defined as follows: for each w ∈ T_xM, m_x,y(w) is obtained by first parallel transporting w among γ_x,y onto T_yM and then reflecting the resulting vector with respect to the hyperplane perpendicular to the geodesic at y. Recall that for x,y∈Ω we let {e_i}_1≤ i≤ N be an orthonormal basis at x with e_N=γ'_x,y(-d2) and parallel transport {e_i}_1≤ i≤ N along γ_x,y. We then have that m_x,y(e_i) = e_i for i = 1, … , N-1 and m_x,y(e_N) = -e_N.
  For any function G: M× M →ℝ, we denote G̃ :O(M)× O(M)→ℝ to be its lift onto O(M) defined by G̃ = G ∘ (π, π).
  
  In particular, we take (B_t)_t≥ 0 to be a standard ℝ^N-valued Brownian motion and for α∈ [0,1], we construct the α-parametrized family of diffusions on O(M)× O(M) given by
 
    dU_t=∑_i=1^N√(2)H_e_i(U_t)∘ dB^i_t+2H_∇ v(U_t)dt+2α_K(ξ̃(U_t,V_t))γ̃'(U_t)dt,    U_0 = u_x
       
       dV_t=∑_i=1^N√(2)H_e_i(V_t)∘ dW^i_t+2H_∇ v(V_t)dt-2α_K(ξ̃(U_t,V_t))γ̃'(V_t)dt,
        V_0 = m_x,yu_x  
       dW_t=V_tm_X_tY_t(U_t)^-1 dB_t,
       X_t=π(U_t),      Y_t=π(V_t).
 
Here, we denote ξ_t=d(X_t,Y_t)/2 and γ̃ to be the horizontal lift of γ onto O(M). Furthermore, we let u_x be the frame (e_i)_i at x.  


The key ingredient in the stochastic calculus is Itô's formula  (see, e.g., Chapter 4 in <cit.>). However, before doing the main computation, we first recall several lemmas from <cit.>. The first shows that if X_0,Y_0∈Ω, the processes X_t,Y_t will stay in Ω for all t≥ 0.
		
		Let Ω be a bounded strictly convex domain in a Riemannian manifold with smooth boundary ∂Ω.	If X_0, Y_0∈Ω, then X_t,Y_t∈Ω for all t≥ 0.
		 
  
      The proof is nearly identical to Lemma 3.2 of <cit.>. In order to follow their proof, one needs to justify that inf_∂Ω|∇ u_1|>0 which follows from Lemma 3.4 of <cit.>.
  
We then consider the 'coupling time` τ, which is defined to be the smallest time at which the diffusions collide

    τ :=
    inf{t ≥ 0 : X_t = Y_t }.

The second lemma shows that this time is finite almost surely.
  
      For any X_0,Y_0 ∈Ω, we have that τ <∞ almost surely.
  
The proof is identical to the proof of Lemma 3.6 in <cit.>.

 With these two lemmas in hand, we now use Itô's formula to obtain the following proposition.

   For t < τ, we have that 
      
    d ξ_t = √(2)d β_t+ {-(N-1+2α )_K(ξ_t)+F_t}dt,

      where β_t is a one-dimensional Brownian motion and  ξ_t = d(X_t,Y_t)/2. 
      In particular, for any smooth function ϕ, we have that 
      
    dϕ(ξ_t)=√(2)ϕ'(ξ_t)dβ_t+{-(N-1+2α)_K(ξ_t)ϕ'(ξ_t)+ϕ'(ξ_t)F_t+ϕ”(ξ_t) }dt.

  
  
      (<ref>) follows from (<ref>) by the standard Itô's formula. On the other hand, (<ref>) follows from applying Itô's formula to the lift ξ̃_t = ξ_t∘ (π, π) (see also Section 6.6 in <cit.> for a similar computation) and the well-known second variation of distance in 𝕄^N_K
      
    ∑_i = 1^N∇ ^2_E_i,E_iξ(x,y) = -(N-1) _K(ξ(x,y)).

  



















 §.§ Proof of Proposition <ref>

Now that we have defined the relevant stochastic equations, our next goal is to show that the first eigenfunction satisfies a log-concavity estimate. In this section, we use the diffusion (X_t, Y_t) for α =1.
  Let γ_X_t,Y_t be the normal minimal geodesic that goes from X_t to Y_t with γ_X_t,Y_t(-ξ_t) = X_t and γ_X_t,Y_t(ξ_t) = Y_t.
We define 

    F_t= Z(X_t,Y_t).

We first derive the SDE for F_t, which we obtain from Itô's formula.
	
	Let λ = λ_1(Ω, ρ, V) be the first eigenvalue of the problem (<ref>) on a convex domain Ω⊂𝕄^n_K and v=log u_1. Then F_t satisfies
	
    dF_t   =d{}
          +ℱ_∇ [-λρ + V](X_t, Y_t)+2_K(ξ_t)𝒞_[λρ -V](X_t, Y_t) dt 
           +(N-1)(K-^2_K(ξ_t))F_tdt
          +2_K(ξ_t)(⟨∇ v(Y_t), e_N  ⟩^2+⟨∇ v(X_t), e_N ⟩^2)dt
          +2/_K(2ξ_t)∑_i=1^N-1(⟨∇ v(Y_t), e_i  ⟩-⟨∇ v(X_t), e_i  ⟩)^2dt.

	
Here, `martingale' is a martingale which does not need to be specified for the purposes of this proof. However, it can be written out explicitly.

To show this, we let Z̃ be the lift of Z onto O(M) × O(M). In other words, Z satisfies Z̃∘π = Z.  Since (U_t,V_t) solves the SDE  (<ref>), and since dF_t = dZ̃ (U_t,V_t), we find that  
     
    d  F_t    = d {}+ {∑_ i= 1^N ∇^2_E_i,E_iZ(X_t,Y_t)+2∇_∇ v(x) ⊕∇ v(x)Z+∇_E_NZ}dt,

     (see <cit.> Section 6 or <cit.> Proposition 4.1 for a similar computation). Then the claim follows from Proposition <ref>. We are now in the position to prove Proposition <ref>.


We first note that from Equation <ref>, since ρ is a modulus of concavity for ρ and V a modulus of convexity for V, we have that

    dF_t- d{}≥   {-2λ_1ρ'(ξ_t)+4λ_1 _K(ξ_t)min_Ωρ}dt 
       +{2V'(ξ_t)-4 _K(ξ_t)max_Ω V }dt 
        +(N-1)(K-^2_K(ξ_t))F_tdt
        +2_K(ξ_t)(⟨∇ v(Y'_t), e_n  ⟩^2+⟨∇ v(X'_t), e_n  ⟩^2)dt
    ≥   {-2λ_1ρ'(ξ_t)+4λ_1 _K(ξ_t)min_Ωρ}dt 
       +{2V'(ξ_t)-4 _K(ξ_t)max_Ω V }dt 
        +{(N-1)[K-^2_K(ξ_t)]+_K(ξ_t)F_t}F_tdt,

where we used the inequality
 
    ⟨∇ v(Y_t), e_N  ⟩^2+⟨∇ v(X_t), e_N  ⟩^2 ≥(⟨∇ v(Y_t), e_N  ⟩ - ⟨∇ v(X_t), e_N  ⟩)^2 /2 = F_t^2/2
 
in the second inequality above. 
On the other hand, in view of Proposition <ref>, we get that 

    d{(2ψ(ξ_t)+(N-1)_K(ξ_t)}-d{martingale}
     
    =   {-2(N+1)_K(ξ_t)ψ'+2ψ”-(N-1)^2[K+_K^2(ξ_t)]_K(ξ_t)+ ((N-1)(K+_K^2(ξ_t))+2ψ')F_t }dt.
				
Combining (<ref>) and (<ref>) using (<ref>), we have that

    d{F_t-2ψ(ξ_t)-(N-1)_K(ξ_t)}-d{}
    ≥   {-2λ_1ρ'(ξ_t)+4λ_1 _K(ξ_t)min_Ωρ +2V'(ξ_t)-4 _K(ξ_t)max_Ω V}dt 
       +{ 2(N+1)_K(ξ_t)ψ'-2ψ”+(N-1)^2[K+_K^2(ξ_t)]_K(ξ_t)}dt 
        +{-2(N-1)^2_K(ξ_t)-2ψ'(ξ_t)+_K(ξ_t)F_t}F_tdt
    ≥   { 2(N-1)_K(ξ_t)ψ'+4ψψ'-4_K(ξ_t)ψ^2+(N-1)^2_K^3(ξ_t)}dt 
        +{-2(N-1)^2_K(ξ_t)-2ψ'(ξ_t)+_K(ξ_t)F_t}F_tdt
    ≥   { 2(N-1)_K(ξ_t)ψ'+4ψψ'-4_K(ξ_t)ψ^2+(N-1)^2_K^3(ξ_t)}dt 
       +{-2(N-1)^2_K(ξ_t)-2ψ'(ξ_t)+_K(ξ_t)F_t}(2ψ +(N-1)_K(ξ_t))dt
        +{-2(N-1)^2_K(ξ_t)-2ψ'(ξ_t)+_K(ξ_t)F_t}(F_t-2ψ -(N-1)_K(ξ_t))dt
    
      
    	
     
     
    	
     
      
    	
     =   {-(N-1)^2_K(ξ_t)-2ψ'(ξ_t)+_K(ξ_t)(F_t+2ψ)}(F_t-2ψ -(N-1)_K(ξ_t))dt

For convenience, we write Ψ = ψ +N-12_K 

    d{ e^∫_0^t-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds(F_t-2Ψ(ξ_t))  }
       ≥ e^∫_0^t-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)dsd{martingale}.

Letting T>0 be some positive cut-off time, we integrate from 0 to  min{τ, T } (denoted τ∧ T), which gives 

    ( e^∫_0^τ∧ T-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds(F_τ∧ T-2Ψ(ξ_τ∧ T)  )≥(F_0-2Ψ(ξ_0))+Martingale_τ∧ T.

Since τ∧ T is bounded, the stopped martingale `Martingale_τ∧ T' in the above inequality is another martingale. Therefore, we can take the expected value to obtain

    F_0-2Ψ(ξ_0)   ≤𝔼( e^∫_0^τ∧ T-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds(F_τ∧ T-2Ψ(ξ_τ'∧ T)  )
       ≤𝔼( e^∫_0^τ∧ T-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds|F_τ∧ T-2Ψ(ξ_τ∧ T)|).

By Fatou's Lemma, we have that

    F_0-2Ψ(ξ_0)≤𝔼( lim inf_T→∞ e^∫_0^τ∧ T-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds|F_τ∧ T-2Ψ(ξ_τ∧ T)|).

Since τ<∞ almost surely, we find that τ∧ T→τ as T→∞. Since  F_τ=Ψ(ξ_τ)=0 and since τ<∞ a.s.

    𝔼( lim inf_T→∞ e^∫_0^τ∧ T-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds|F_τ∧ T-2Ψ(ξ_τ∧ T)| )
       =𝔼(  e^∫_0^τ-(N-1)^2_K(ξ_s)-2ψ'(ξ_s)+_K(ξ_s)(F_s+2ψ)ds|F_τ-2Ψ(ξ_τ)| )=0.

We conclude that

    F_0-2ψ(ξ_0)-(N-1)_K(ξ_0)≤ 0,

as desired.



  §.§.§ Sketch of Maximum Principle Proof



From Proposition <ref>, we can understand the behaviour of Z at a maximal point. In particular, it is straightforward to obtain the following theorem. We simply sketch a proof here.

    Under the same assumptions as in Theorem <ref>, assume that ψ_0+N-12_K: [0,D/2] → is a modulus of concavity of v. Then if ψ is a smooth solution to the problem 
    
    ∂ψ∂ t≥ψ” + 2 ψ'ψ +λρ' - V'-2_K(s) ( ψ'+ψ^2+ρ-V)
    ψ(t,0) = 0
    ψ(0, ·) = ψ_0,

    then we have that ℱ_∇ v(x,y) ≤ 2 ψ(t, d2)+(N-1)_K( d2) for all x,y ∈Ω and t ≥ 0.


    Consider Z_ε (x,y,t) = ℱ_∇ v(x,y) - 2ψ(t,  d2)-ε e^Ct and argue by contradiction, assuming there is a (x_0,y_0,t_0) such that the maximum of Z_ε is equal to zero.  Then at that point all first order derivatives vanish and the proof is exactly analogous to the one in <cit.>.  




 §.§ Spectral Gap Comparison

We now prove a spectral gap comparison. To do so, we use the diffusions (X_t, Y_t) from (<ref>) with α = 0.
   
       Suppose that Ω⊂ (M, g_𝕄^N_K) is convex and suppose that 
       
    ℱ_∇ v(x,y)≤ 2(logφ_1)' ( d(x,y)2)+(N-1)_K(d(x,y)2).

       Then we have that the fundamental gap of the problem (<ref>) satisfies 
       
    Γ(Ω) ≥minρ/max_ΩρΓ(ρ, V).

   

Note that

    Γ(Ω, ρ)   =min _V⊂ H^1,  V=2max _f ∈ V∫_Ω|∇ f|^2 u^2  dx/∫_Ωρ f^2 u^2  dx
       ≥1/ρ_∞min _V⊂ H^1,  V=2max _f ∈ V∫_Ω|∇ f|^2 u^2  dx/∫_Ω  f^2 u^2  dx
       = μ/ρ_∞.

Let φ_1,φ_2 be the first two eigenfunctions of the corresponding one-dimensional model (<ref>). Define Φ=φ_2/φ_1 then we have by Proposition <ref> 

    dΦ(ξ_t)=√(2)Φ'(ξ_t)dβ_t+{-(N-1)_K(ξ_t)Φ'(ξ_t)+Φ'(ξ_t)F_t+Φ”(ξ_t) }dt.

By our assumption, we have F_t = ℱ_∇ v(X_t,Y_t)≤ 2(logφ_1)'(ξ_t)+(N-1) _K(ξ_t). 

    dΦ(ξ_t)≤ d{}+{Φ'(ξ_t)2(logφ_1)'(ξ_t)+Φ”(ξ_t) }dt.

Moreover, straightforward computation gives 

    Φ”(s)+Φ'(s)2(logφ_1)'(s)=-Γρ(s)Φ(s).
 
Combining (<ref>) and (<ref>) gives

    dΦ(ξ_t)≤ d{}-ΓρΦ(ξ_t)dt,

which is equivalent to 

    d(e^∫_0^tΓρΦ(ξ_t) )≤ d{}.

Integrating and taking expectation gives

    𝔼(e^∫_0^tΓρΦ(ξ_t) )≤Φ(ξ_0),

which gives

    𝔼Φ(ξ_t)≤ e^-minρΓtΦ(ξ_0).

By Lemma <ref>, we know that the above inequality holds for all t≥ 0. Finally, denote w(x) to be the Neumann eigenfunction to μ and letting w(x,t)=e^-μtw(x) be the solution to the associated heat flow. Then, from the Feynman-Kac formula, we have w(t,x)=𝔼[w(X_t)]. Since w is Lipschitz on Ω̅, we can find Λ>0 such that
				
    |w(x,t)-w(y,t)|   =|𝔼[w(X_t)-w(Y_t)]|≤𝔼|w(X_t)-w(Y_t)|
       ≤Λ𝔼d(X_t,Y_t)=2Λ𝔼ξ_t.

As before, Φ'(0)>0; from which we deduce that there exists c_1>0, such that Φ(s)≥ c_1 s on [0,D/2]. Hence, we have from (<ref>), that
				
    c_1𝔼ξ_t≤𝔼(Φ(ξ_t))≤ e^-ΓminρtΦ(ξ_0).
 
				Putting this together with (<ref>) and the definition of w gives
				
    e^-μt |w(x)-w(y)|=|w(x,t)-w(y,t)|≤2Λ/c_1e^-ΓminρtΦ(ξ_0).

				Since this must hold for all t>0, taking the limit as t gets large gives the claim.
			


§ THE ONE-DIMENSIONAL MODEL



In this section, we study the fundamental gap in the case n= 1 in order to get quantitative bounds on the fundamental gap. In other words, we let ρ to be a smooth and uniformly positive function, and consider the problem (<ref>):

    -φ”+ Vφ= λρφ  [- D2,  D2],     φ(- D2) = φ(  D2) = 0.

Using the Raleigh quotient formulation of the first eigenvalue, it is easy to show the following lemma. 

If min V ≥ -k^2π^2/D^2,
   we have for the k-th eigenvalue of the problem (<ref>)
    
    1/maxρ(k^2π^2/D^2+minV) ≤λ_k ≤1/minρ( k^2π^2/D^2+maxV)


 
    This follows from the Raleigh quotient: to see the first inequality, note that
    
    ∫ (u')^2+Vu^2  ds/∫ρu^2  ds≥1/maxρ(π^2/D^2+minV).

   One now uses the min max characterization of eigenvalues and the result follows. 

We now let L >0 and we consider the problem 

    -φ_L” +Vφ_L-λ_Lφ_L = 0   [-L2, L2]  &  φ(-L2) = φ(L2) = 0,

and we denote φ_L,i to be the i-th eigenfunction of that problem and λ_L to be the i-th eigenvalue. Via a standard Ricatti comparison, one immediately gets the following proposition (for vanishing potential V≡ 0).

Let ψ = (logφ_1)' and ψ_L = (logφ_L,1)' both with V≡ 0. Then for any L ≥√(maxρ)√(minρ)D, we have that 

    ψ≤ψ_L  [-D/2, D/2].


Note that ψ and ψ_L satisfy both a Ricatti equation:
       
    ψ' + ψ^2 + λρ   = 0
    ψ_L' +ψ_L^2+λ_L     = 0,

with initial condition ψ_L(0) = ψ(0) = 0.
By our choice of L, we have that 

    λρ≥λ_L = π^2/L^2.

Applying a Ricatti comparison, we conclude

    ψ≤ψ_L.


The following result provides a lower bound on the fundamental gap.

For any potential V≥ 0, the fundamental gap of the problem (<ref>) satisfies 
    Γ(ρ, V)≥3π^2/D^2minρ/(maxρ)^2-(maxV/minρ- minV/maxρ)




To see this, note that 
    minV/maxρ+λ_i(ρ, 0) ≤λ_i(ρ, V) ≤maxV/minρ +λ_i(ρ, 0).

Hence,

    Γ(ρ, V) ≥Γ( ρ, 0)-(maxV/minρ- minV/maxρ).

To estimate the gap Γ(ρ) : = Γ(ρ, 0), we make use of the fact that w = φ_2φ_1 satisfies

    w”+2 (logφ_1)'w ' = - Γ(ρ)ρw  [- D2,  D2],   w'(- D2) = w' ( D2) = 0.

Hence we estimate the first Neumann eigenvalue of the operator -d^2ds^2+2(logφ_1)'dds. To do so, observe that 

    Γ(ρ)    = min_V:  (V) = 2max_ϕ∈ V∫_- D2^ D2(ϕ')^2φ_1^2  ds/∫_- D2^ D2ρϕ^2φ_1^2  ds
       ≥1/maxρmin_V:  (V) = 2max_ϕ∈ V∫_- D2^ D2(ϕ')^2φ_1^2  ds/∫_- D2^ D2ϕ^2φ_1^2  ds
        = 1/maxρμ,

where μ is the first non trivial Neumann eigenvalue of the problem 
 
    w”+2 (logφ_1)'w ' = -μw  [- D2,  D2],   w'(- D2) = w' ( D2) = 0.

 We now set ψ = (logφ_1) ' and ψ _L = (logφ_L,1)', where φ_L,1 is the first eigenfunction of the problem (<ref>). We then see from Proposition <ref> that

    ψ≤ψ_L.

The claim then follows from similar arguments as in Theorem <ref>. Alternatively, it follows directly from Proposition 3.2 in <cit.>. 



§ HOROCONVEX DOMAINS AND THEIR FUNDAMENTAL GAP




    
We consider the Poincaré model

    g_ℍ^N = 4/(1-x^2)^2g_ℝ^N.

In view of (<ref>), we have 

    Γ(Ω, Δ_g_ℍ^N) = Γ(Ω, Δ_ℝ^N, e^2φ)

where 
    exp(2φ) = 4/(1-x^2)^2,     φ = log(2/1-x^2).

To apply Theorem <ref>, we need to construct a modulus a modulus of concavity for ρ = exp (2 φ).
We calculate the eigenvalues of the Hessian of ρ are given (in polar coordinates)
 
    σ^ρ_1 =   16/(1-r^2)^3,   σ^ρ_2 =  16(1+5r^2)/(1-r^2)^4.

    Let R denote the circumradius in hyperbolic geometry. Then, we denote R_E = tanh(R2) its Euclidean radius.
   We have that 
    ρ_C (s) = sup_Ωmax{σ^ρ_1, σ^ρ_2}s^2/2+C=(max_Ωρ)^2(1+5R_E^2)s^2/2 +C

is a modulus of concavity for any 
constant C>0. In particular, 

    C ≤ρ≤34(D_Emax _Ωρ)^2 +C,

where D_E denotes the diameter of Ω with respect to the Euclidean metric. 
Since ρ' >0, we need
that the eigenvalue of the problem (<ref>) with ρ= ρ_C satisfy the condition (<ref>).
To ensure this, we fix a number C>0, and choose 

    V≡max_s∈ [0,D/2]ρ(s) (λ_1(Ω, Δ_ℝ^N, e^2φ) -λ(ρ, 0)).

To make the choice of V depend only on the hyperbolic diameter of Ω, note that

    D_E ≤D_ℍ^n/2,

as one can see by computing length of curves and taking infimums. To remove the dependence on λ_1(Ω, Δ_ℝ^N, ρ), we recall that for ρ= e^2φ

    λ(Ω,Δ_ℝ^N, ρ) = λ_1(Ω, Δ_ℍ^n)-N(N-2)/4≤λ_1(B_r, Δ_ℍ^n)-N(N-2)/4,

where B_r⊂Ω is the largest ball inscribed in Ω.


For horoconvex domains, Theorem 1 in <cit.> shows that the inradius will be bounded from below by the diameter. More precisely, we have that

    D/2≤ξ(r) +r,

where 
    ξ(r) =  ln((1+√(τ))^2/1+τ)    τ(r) = tanh ( r2).

To find the lower bound, note that ξ(r) ≤ 2 √(r) and thus  

    r ≥(-1+√( D2+1))^2.

One now employs an upper bound for the first Dirichlet eigenvalue of balls in hyperbolic space (see e.g. Theorem 5.6 in <cit.>) to find that 

    λ_1(B_r, Δ_ℍ^N) ≤(N-1)^2/4+ π^2/r^2+ C/r^3,

where C= π^4(N^2-1)12.
Putting everything together, we obtain that 

    λ_1(Ω, Δ_ℝ^N, ρ) ≤1/4+π^2/(-1+√( D2+1))^4+ (N^2-1)π^4/12(-1+√( D2+1))^6.

Hence we can choose V to be greater or equal than a constant, depending only on the diameter of Ω.
Then, all the assumptions of Theorem <ref> are satisfied and we conclude that 

    ℱ_∇ v(x,y) ≤ 2(logφ_1)'( d2)

By Theorem <ref>, we conclude that 

    Γ(Ω)   ≥minρ/max_ΩρΓ(ρ, V).





 §.§ Explicit bounds on the gap

In this subsection, we prove closed-form estimates on the spectral gap. The idea is to use the bound of Proposition <ref> to derive a bound on the Hessian and then employ Theorem 3 of <cit.>. From Proposition <ref>, we can divide (<ref>) by d and passing to the limit as d → 0^+. Doing so, we find that

    ∇^2 v ≤ψ'(0),

where ψ = (logφ_1)'. Since ρ is even, we have that ψ is odd and hence we get that 

    ψ'(0) = -ψ^2(0) -λρ(0)+ V = -λ(ρ, V) minρ+ V.

To estimate the right-hand side of the above inequality, note that by Theorem <ref>

    -λ(ρ, V) minρ+ V   ≤ -λ(ρ,0)minρ+V(1-minρ/maxρ) 
       ≤ -λ(ρ,0)maxρ+ 3/4(D_E^2max _Ωρ)^2 λ_1(Ω, Δ_ℝ^N, ρ).

We now estimate 3/4(D_E^2max _Ωρ)^2 λ_1(Ω, Δ_ℝ^N, ρ) in terms of the diameter of Ω. Note that max _Ωρ = 4/(1-R_E^2)^2, where R_E is the Euclidean circumradius of Ω. A result by Dekster <cit.> shows that

    R_ℍ^N≤(√(2N)/√(N+1)sinh(D)),

 where we shorten D_ℍ^N to D.

From this, we find that

    R_E ≤tanh(1/2(√(2N)/√(N+1)sinh(D)))

and further estimate

    1/(1-R_E^2)   ≤cosh^2(12(√(2N)√(N+1)sinh(D)))
       ≤cosh^2((√(2N)√(N+1)sinh(D)))
        =1+ 2NN+1sinh^2(D).

We thus get that 

    34(D_E max_Ωρ) ^2 ≤ 3(4 ∧ D^2)( 1+ 2NN+1sinh^2(D))^4,

 where here and in the following we denote a∧ b: = min{a,b} for any a,b∈ℝ. 

Lastly, we use the estimate from Theorem <ref> and find that -λ(ρ) maxρ≤ -π^2/D_E^2≤ -π^2(4/D^2∧ 1) (we again use the notation ∧ to indicate the maximum)
Putting these together, we find that 

    ∇^2 v ≤ -π^2 (1 ∧4/D^2)+ 3(4 ∧ D^2)( 1+ 2NN+1sinh^2(D))^4( 1/4+π^2/(-1+√( D2+1))^4+ (N^2-1)π^4/12(-1+√( D2+1))^6).

This upper bound is only in terms of the diameter and the dimension, so we denote the right hand side of the inequality as R(N, D). We finally conclude that 

    Γ(Ω, Δ_ℍ^N)    =
       min _V⊂ H^1,  V=2max _u ∈ V∫_Ω|∇ u|^2 e^-f  dx/∫_Ωρ u^2 e^-f  dx
       ≥1/ρ_∞min _V⊂ H^1,  V=2max _u ∈ V∫_Ω|∇ u|^2 e^-f  dx/∫_Ω u^2 e^-f  dx  = μ/ρ_∞.

The Neumann eigenvalue μ can now be estimated using Theorem 3 of <cit.> and we conclude that 

    Γ(Ω) ≥π^2/(4∧D^22)(1+ 2NN+1sinh^2(D))^2exp(-C_N(1∧ D2)√(R(N,D))).




  §.§.§ Asymptotic bounds on the gap




For D ≫ N >2, it is possible to rewrite this inequality completely explicitly. In particular, using the derivation from Theorem 3 of <cit.>, after some algebraic manipulations we obtain the asymptotic estimate 

    Γ(Ω) >  π^2(N-1)^2 D^2/16exp[ - (N-1) (D^2) (1+ 2 exp(2D) )^2  ].
 

From this, we obtain a bound which decays at doubly-exponential rate in terms of the diameter. Nguyen, Stancu and Wei <cit.> showed that the gap of large horoconvex domains is bounded from above by C_N/D^3, so these two results raise the following question.

    How quickly does the fundamental gap of a horoconvex domain decay in terms of its diameter?


The gap estimate we obtain in (<ref>) also decays when D is small, since in this case the final term in R(N,D) is dominant and tends to infinity. This might be somewhat unexpected, since the fundamental gap conjecture posited that in ℝ^N, Γ(Ω) ≥3 π^2/D^2. We will provide an estimate for small horoconvex domains in Section <ref> which is asympotically of this size. However, let us explain the reason for why the gap estimate in the previous section decays as the diameter does.

Although our focus in this section has been horoconvex domains, the modulus of concavity estimate applies to any domain which is convex with respect to the Poincaré disk model. In particular, the entire argument (except for Equation (<ref>)) applies to thin rectangles in the disk model. In other words, such domains have the Hessian upper bound: 

    ∇ v^2≤ -π^2 (1 ∧4/D^2)+ 3(4 ∧ D^2)( 1+ 2NN+1sinh^2(D))^4λ_1(Ω, Δ_g_, e^2φ)

which blows up for thin rectangles. 

For such domains, one can essentially replicate the argument of <cit.> to show that the fundamental gap can go to zero as the inradius tends to zero (see Appendix <ref> for details). As such, it is not possible to obtain a uniform estimate for  the fundamental gap for domains which are convex in the Poincaré disk model solely in terms of the diameter and the dimension. In particular, the estimate must incorporate the inradius (or equivalently, the diameter). As the diameter goes to zero, the inradius must also go to zero. Therefore, the estimate we obtain must tend to zero as the diameter shrinks in order to give a valid theorem. Therefore, any proof which obtains more refined bounds of the gap of horoconvex domains can only apply to a more restrictive class of domains.




The following comment just derives the asymptotic bound as the diameter of a horoconvex domain gets large 
The bound from <cit.> is the following.
  
    μ_1    ≥   π^2/16·max (n-1,2)(n-1) k/(exp(1 / 2 max (√(n-1), √(2)) √((n-1) k d^2))-1)^2
        =    π^2/16 (n-1)^2 k/(exp(1 / 2 (n-1) √(k) d )-1)^2
       ≥   π^2/16 (n-1)^2 k/exp( (n-1) √(k) d  )

  Meanwhile, the estimate on the Hessian of v shows that for D large, 
  
    k < D^2 (1+2 exp(2D))^4

where I'm using 1>3/4 to absorb some of the lower order terms. Furthermore, the computation on R_E shows that ρ≤ 4(1+ 2 exp(2D))^2.

Therefore, we have that

    μ/ρ≥π^2/16(N-1)^2 D^2/exp((N-1) D^2 (1+ 2 exp(2D))^2 )

Doing one more large D simplification, we have that
  
    μ/ρ≥π^2/16(N-1)^2 D^2/exp((4+o(1))(N-1) D^2  exp(4D) )









§ FUNDAMENTAL GAP ESTIMATES ON CONFORMALLY FLAT MANIFOLDS


The approach we presented in Section <ref> also works for more general conformally flat manifolds. In particular, Proposition <ref> shows that for any domain which is convex with respect to the flat connection, we have that Ω⊂ (M, g = e^2φg_), that 

    Γ(Ω, Δ_g) ≥Γ(Ω, Δ_g_, e^2φ)-N-2/4(N-1)_Ω(R_g).

We are now in the position to present the 

    To estimate Γ(Ω, Δ_g_, e^2φ) we let ρ be a modulus of concavity for ρ = e^2φ with respect to g_. In other words, we let σ_max = (max_Ωσ(∇^2 ρ)) be the largest eigenvalue of the Hessian of ρ with respect to the Euclidean metric. If σ_max is negative (i.e., ρ is concave), we replace σ_max by 0, which will ensure that ρ is constant and that <ref> holds automatically. In other words, σ_max := (max_Ωσ(∇^2 ρ) ∨ 0).
    We then set 
    
    ρ(s) = σ_maxs^22+C

    for some constant C>0.
    We then choose V as in Section <ref> and the whole proof goes through so long as 
    
    Γ(Ω, Δ_g) ≥minρ/max_Ωexp(2φ)Γ(ρ, V)-N-2/4(N-1)_Ω(R_g).



Note that when σ_max is positive, the quantity V depends on λ_1(Ω), and this dependence on is crucial since if the domain collapses to a line, the gap can become arbitrarily small (see Appendix <ref> for details).  However, if one restrict to domains Ω whose inradius r_Ω is bounded from below by a positive function of the diameter D, one can make the estimate to be dependent only on the diameter of Ω and the conformal factor.






   Suppose that Ω is convex with respect to g_ and whose  inradius is bounded below by some positive function of the diameter. Then there exist ρ and V depending only on the conformal factor and D such that (<ref>) holds.


When ρ is concave, we obtain a gap estimate which is independent of the inradius (c.f. Theorem 4.1 of <cit.>). In particular, we obtain the estimate
    
    Γ(Ω) ≥minexp(2φ)/maxexp (2φ)3π^2/D^2-(N-2)/4(N-1)_Ω(R_g).

    


  §.§.§ A worked example: 𝕊^1 ×𝕊^N-1



As a demonstration of how this theorem can be applied, we consider M = 𝕊^1×𝕊^N with its standard metric, which is conformally-flat. When N is even, we can also consider Hopf manifolds. 

For domains Ω⊂𝕊^1 ×𝕊^N-1, it will be convenient to consider the affine universal cover ℝ^N \ 0 with the conformal metric g_𝕊^1 ×𝕊^N-1 = 1/r^2 g_, where r = x^2.  
When N is even, we can consider ℝ^N \ 0 as ℂ^N/2\ 0 and construct Hopf manifolds as the quotient of this cover by the action of ℤ generated by an holomorphic contraction z →α z where α∈ℂ satisfies |α|<1. The class of domains Ω⊂ℝ^N \ 0 that we consider are those which are convex and compact with respect to the affine structure.

Then, a straightforward computation shows that the eigenvalues of 
the Hessian of 1/r^2 are -2/r^4, repeated with multiplicity N-1 and 6/r^4 with multiplicity 1. Furthermore, since 𝕊^1 ×𝕊^N-1 has constant scalar curvature, bounding the gap reduces to bounding the gap of (<ref>).
As before, we construct a modulus of concavity ρ given by 
    
    ρ(s) = 6/inf_x ∈Ω r^4s^22+C,

    where C is a positive constant which we can take to be inf_x ∈Ω1/r^2 

    In order to obtain a gap estimate, we must verify that λ(ρ, V) ≥λ_1(Ω,  ρ), for ρ = 1r^2. To do so, we choose V large enough so that the desired eigenvalue comparison holds. Doing so, we will get an upper-bound on the log-concavity of u_1, and thus a lower bound on the fundamental gap of the domain in terms
    of the diameter and the principle eigenvalue of the domain.
    
    
    From this, we find that Theorem <ref> that
    
    Γ(Ω, ρ) ≥minρ/max_ΩρΓ(ρ, V).

        




 §.§ Gap estimates for small horoconvex domains




In <cit.>, we considered domains which are convex with respect to a particular spherical geometry, which is a more restrictive class of domains, though general enough to include horoconvex domains up to a certain diameter. We can again use positively curved geometry as the reference, and doing so well yield stronger gap estimates compared to our earlier work when the diameter is small.

To do this, we again consider the Poincaré disk model (ℍ^N, g_ℍ^N) = (B_1(0), 4(1-r^2)^2g_) for the hyperbolic space and consider a sphere of radius R (i.e. K = 1/R^2). Using stereographic projection map from the disk model to the sphere, we can relate the hyperbolic metric to a spherical one by the conformal change g_ℍ^N = (R^2+r^2)^2R^4(1-r^2)^2g_𝕊^N_K on the ball B_R(0) ⊂ B_1(0). For diameters D small enough, one can repeat the argument of the proof of Theorem <ref> to obtain the estimate
  
    Γ(Ω) ≥(min_Ωρ-σ_maxD^28)^3/min_Ωρ(max_Ωρ)^23π^2/D^2-N(N-2)/4_Ω(ρ^-1),

    where ρ = exp(2φ)=(R^2+x^2)^2R^4(1-x^2)^2 and σ_max chosen as in (<ref>). By taking the diameter of Ω small and choosing the radius R>0 small enough, the modulus of concavity ρ of ρ will satisfy the assumptions of Theorem <ref>.
In particular, we can construct a modulus of concavity ρ of ρ given by 
    
    ρ(s) = σ_maxs^22+C.

  In order to apply the comparison, we can choose D>0 small enough and K>0 large enough (i.e. R small enough) such that for C = min _Ωρ-D^28σ_max>0, ρ satisfies 
    
    ρ≤minexp(2φ)   &  ρ' ≤ 2_K ρ.

Thus (<ref>) follows.

    





 §.§ Spectral Gap Estimates on Surfaces of Positive Curvature




In <cit.>, Andrews posed the following question.

[Page 19 <cit.>]
   [C]an one expect a useful lower bound on the gap on
a strongly convex Riemannian manifold by controlling curvature, or is it necessary to control
higher derivatives of curvature as well?


Since this question was first posed, there have been a number of developments. For instance, the curvature must be non-negative in order to establish gap estimates for Levi-Civita convex domains. However, the situation in higher-dimensional manifolds remains open, even for well-understood spaces like ℂℙ^n (see <cit.> for some recent progress on this topic).

Nonetheless, it is possible to obtain gap estimates for surfaces whose curvature is positive and controlled in a C^2 sense (see Theorem 1.1 of <cit.>). But this leaves unresolved the question of whether it is possible to obtain gap estimates when the curvature is simply positive. Although it involves a different notion of “convexity" for domains, the results in this paper give evidence that positive curvature suffices to establish gap estimates.

Indeed, suppose that (M^2,g) is a surface of Gaussian curvature κ. From the uniformization theorem, we can locally write g = e^2φg_ where φ is a function satisfying (<ref>), which simplifies to 
    κ=-2(N-1) e^-2φΔ_φ.
 If e^2φ is concave with respect to g_, i.e. if 

    φ +2dφ⊗ dφ≤ 0,
 then we can let ρ≡ 1 which is a modulus of concavity for ρ = e^2φ. In that case, for any Ω⊂ M that is convex with respect to g_, we find that 

    Γ(Ω) ≥minexp(2φ)/maxexp(2φ)3π^2/D^2.

So we have that whenever the conformal factor is concave, one can prove strong gap estimates. Note that the curvature could be positive even if the conformal factor is not concave (but Δ_φ <0). For these situations, our approach leads us to prove lower bounds on the fundamental gap involving the first eigenvalue (as above) but not solely the diameter.






Note that this estimate differs from the estimate given in <cit.>. In fact, there the authors along with Wei showed (among other things) that there is α= α(κ,κ) >0 such that for any surface (M^2,g) whose sectional curvature κ>0, satisfies κ≤κ≤κ and that |Δκ|_∞, |∇κ|_∞≤α(κ, κ) then for any g convex set Ω of diameter D ≤π2√(κ) we have 

    Γ(Ω) ≥3π^2/D^2-(12+3π)(κ-κ).

The assumption of the latter estimate is a C^2 assumption on κ and hence a C^4 assumption on g. On the contrary, the estimate (<ref>) we provide here, does only need the conformal factor to be concave. On the other hand, in Theorem <ref>, we also prove such an estimate for small deformations of spheres in dimension 2, only assuming a C^2 condition on g.
    


    


 §.§ Fundamental gaps on conformal deformations of round spheres

Another application of Proposition <ref> is to obtain fundamental gap estimates of domains in small conformal deformations of round spheres (c.f. Corollary 1.4 of <cit.>[The primary difference between the estimates in our earlier work is that the leading term in the lower bound here is roughly three times as large.]). 

In this section we study the case where M is the sphere 𝕊^N_K of constant sectional curvature K >0 and Ω⊂𝕊^N_K is convex with respect to the round geometry. Then we consider the conformal metric g = e^2φg_𝕊^N_K. As before, we let 

    σ_max = max{ 0, maxσ (  _g_𝕊^N_Ke^2φ)}

which is either 0 if e^2φ is concave or the largest eigenvalue of the hessian of e^2φ, otherwise.

    There exists ε = ε(N,K)>0 such that whenever σ_max< ε, one has that 
    
    Γ(Ω) ≥(min_Ωρ-σ_maxD^28)^3/min_Ωρ(max_Ωρ)^23π^2/D^2-(N-2)/4(N-1)_Ω (R_g)-N(N-2)/4_Ω (exp(-2φ)).



    Let u be the g-eigenfunction on Ω. Then, applying a conformal change procedure as before, we have that ue^N-22φ satisfies the Schrödinger equation (<ref>), where   V = - N-2/4(N-1)e^2φR_g̃+N-2/4(N-1)R_g. Thus, from the Raleigh quotient, we get that 
    
    Γ(Ω, Δ_g) ≥Γ(Ω, Δ_g_𝕊^N_K, e^2φ)-(N-2)/4(N-1)_Ω (R_g)-N(N-2)/4_Ω(exp(-2φ)).

    Therefore, it suffices to estimate Γ(Ω, Δ_g_𝕊^N_K, e^2φ). As before, we construct a modulus of concavity ρ given by 
    
    ρ(s) = σ_maxs^22+C,

    where σ_max is chosen as in the proof of Theorem <ref>. Note that in order to apply the comparison, we can choose ε>0 small enough such that for C = min _Ωρ-D^28σ_max>0, ρ satisfies 
    
    ρ≤minexp(2φ)   &  ρ' ≤ 2_K ρ.

   To finish the proof, we need to verify that λ(ρ) ≤λ_1(Ω,  ρ), for ρ = e^2φ. To do so, note that 
    
    λ_1(Ω, e^2φ) = inf _f ∈ H^1_0(Ω) ∫_Ω |∇ f|^2  dV_g_𝕊^N_K/∫_Ωρ f^2  dV_g_𝕊^N_K≥1/max _Ωρλ_1(Ω) > 1/max_ Ωρπ^2/D_𝕊^N_K^2,

    by Ling's estimate <cit.>, where D_𝕊^N_K denotes the diameter with respect to g_𝕊^N_K. We now consider the problem (<ref>) on the interval [-L/2,L/2] where we choose L = √(max_Ω)ρ√(minρ)D_𝕊^N_K, so that 
    
    λ(ρ, L) ≤1/minρπ^2/L^2≤λ_1(Ω, ρ).

    We thus conclude from Proposition <ref> that (logφ_1)'+N-12_K is a modulus of concavity. Thus by Theorem <ref>, we conclude that 
    
    Γ(Ω, Δ_g_𝕊^N_K, ρ) ≥minρ/max_ΩρΓ(ρ, L).

    In view of Theorem <ref>, we conclude that 
    
    Γ(Ω, Δ_g_𝕊^N_K, ρ) ≥(minρ)^2/(maxρ)^21/max_Ωρ3π^2/L^2 =  (minρ)^3/(maxρ)^21/(max_Ωρ)^23π^2/D_𝕊^N_K^2.

    Finally, since  D_𝕊^N_K≤  D/minexp(φ), we conclude 
    
    Γ(Ω, Δ_g_𝕊^N_K, ρ) ≥(min_Ωρ-σ_maxD^28)^3/min_Ωρ(max_Ωρ)^23π^2/D^2.





alpha






§ RECTANGLES IN THE DISK MODEL OF HYPERBOLIC SPACE





In this appendix, we construct domains in the Poincaré disk model of hyperbolic space which are convex with respect to this geometry but have arbitrarily small fundamental gap. This shows that our assumption that there is a lower bound on the inradius is necessary.


We define the domain Ω to be the Euclidean rectangle with vertices (± L, ± r) in the disk. Here, L is some fixed value and r will be very small.

For the most part, the proof is the same as the proof of Theorem 1 in <cit.>, with two differences.

    
  * We can construct the domain to be symmetric, which eliminates the need for a continuity family at the end of the proof.
    
  * In order to create the neck, we cannot use the growth of Jacobi fields since the domain is convex with respect to the Euclidean geometry, not the underlying hyperbolic geometry. However, we will make use of the fact that the conformal factor exceeds one near the left and right hand sides of the rectangle to replicate this effect.



For r small, we consider three smaller rectangles contained in Ω.

    Ω_Left   =   { (x,y)  |  -L<x<-L/2,  -r<y<r}
    Ω_Neck   =   { (x,y)  |  -L/4<x<L/4,  -r<y<r}
    Ω_Right   =   { (x,y)  |  L/2<x<L,  -r<y<r}


Now, we compute the heights of each of these rectangles with respect to the underlying hyperbolic metric. In other words, we consider the distance from the top of each rectangle to the bottom. Note that this will not be a constant value since the metric is non-Euclidean and will depend on the x-value. Then, a computation of the hyperbolic distances shows that for all points in Ω_Left and Ω_Right, the height of the domain exceeds arcsinh( r/4-L^2-4r^2). Meanwhile, the height of points in Ω_Neck does not exceed
arcsinh(32r/16-L^2-16r^2). We then consider the ratio of these bounds to find the following:

    height(Ω_Left) >  arcsinh( r/4-L^2-4r^2)/arcsinh(32r/16-L^2-16r^2)height(Ω_Neck).


Taking the limit as r goes to zero, L'Hópital's rule shows that

    arcsinh( r/4-L^2-4r^2)/arcsinh(32r/16-L^2-16r^2)→ 16- L^2/16-4L^2.

Therefore, we can find r small enough so that

    height(Ω_Left) >   16- L^2/16-2L^2height(Ω_Neck).

Fixing L positive, 16- L^2/16-2L^2 >1, which suffices to repeat the proof from <cit.> verbatim. In particular, we can bound the eigenvalue of Ω from below by the eigenvalue of Ω_Left, which is at most arcsinh^-2( r/4-L^2-4r^2) +l.o.t.. On the other hand,  the eigenvalue of Ω_Neck exceeds arcsinh^-2(32r/16-L^2-16r^2), so the eigenvalue of the entire domain is much smaller (by a multiplicative factor) compared to the eigenvalue of the middle of the domain. Repeating the same doubling estimate from <cit.>, this forces the principle eigenfunction to become very small through Ω_Neck (roughly O(exp(-cr^-1)), and we can multiply the principle eigenfunction by a cut-off function to obtain a second function whose Rayleigh quotient is very similar to λ_1. Furthermore, since the domain and geometry are symmetric, so long as we choose this cut-off function to be odd, the resulting function is automatically orthogonal to the principle eigenfunction (eliminating the need for a continuity argument). The difference between the Rayleigh quotients of these two functions bounds the fundamental gap, so the fundamental gap of these rectangles gets arbitrarily small as the inradius (i.e. r) goes to zero.

