# Differential Geometry

## New submissions

[ total of 18 entries: 1-18 ]
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### New submissions for Fri, 15 Jan 21


Title: Compact Dupin Hypersurfaces
Authors: Thomas E. Cecil
Subjects: Differential Geometry (math.DG)

A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on $M$, i.e., each continuous principal curvature function has constant multiplicity on $M$. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in ${\bf R}^n$ or $S^n$. The theory of compact proper Dupin hypersurfaces in $S^n$ is closely related to the theory of isoparametric hypersurfaces in $S^n$, and many important results in this field concern relations between these two classes of hypersurfaces. This problem was formulated in 1985 in a conjecture of Cecil and Ryan on p. 184 of the book, "Tight and Taut Immersions of Manifolds," which states that every compact, connected proper Dupin hypersurface $M \subset S^n$ is equivalent to an isoparametric hypersurface in $S^n$ by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.


Title: Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures
Authors: Jun-Muk Hwang
Comments: 20 pages, to appear in J. Math. Soc. Japan
Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Complex Variables (math.CV)

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that ${\mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${\mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$ at some point $x\in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.


Title: A stability result for translating space-like graphs in Lorentz manifolds
Subjects: Differential Geometry (math.DG)

In this paper, we investigate space-like graphs defined over a domain $\Omega\subset M^{n}$ in the Lorentz manifold $M^{n}\times\mathbb{R}$ with the metric $-ds^{2}+\sigma$, where $M^{n}$ is a complete Riemannian $n$-manifold with the metric $\sigma$, $\Omega$ has piecewise smooth boundary, and $\mathbb{R}$ denotes the Euclidean $1$-space. We can prove an interesting stability result for translating space-like graphs in $M^{n}\times\mathbb{R}$ under a conformal transformation.


Title: Lie symmetries of the canonical connection: codimension one abelian nilradical case
Subjects: Differential Geometry (math.DG)

This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to $\nabla$ in the special case where the Lie algebra $\g$ of $G$, has a codimension one abelian nilradical. The conditions that determine a Lie symmetry in such a case are completely integrated. Finally the results obtained are compared with some four-dimensional Lie groups whose Lie algebras have three-dimensional abelian nilradicals, for which the calculations were performed by MAPLE.


Title: Gradient inequality and convergence of the normalized Ricci flow
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)

We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold $\Omega$ without boundary. In \cite{KS10, KS15} we derived, via PDE techniques, global-in-time existence of the classical solution and pre-compactness of the orbit. In this work we show its convergence to steady-states, using a gradient inequality of {\L}ojasiewicz type. We have thus an alternative proof of \cite{ha}, but for general manifold $\Omega$ and not only for unit sphere. As a byproduct of that approach we also derive the rate of convergence according to this steady-sate being either degenerate or non-degenerate as a critical point of a related energy functional.


Title: Equivalences of PDE systems associated to degenerate para-CR Structures: foundational aspects
Authors: Joel Merker (LM-Orsay)
Comments: Supported in part by the GRIEG research project Symmetry, Curvature Reduction, and EquivAlence Methods (SCREAM), 2019/34/H/ST1/00636
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)

Let $K = R$ or $C$. We study basic invariants of submanifolds of solutions $\mathcal{M} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$ in coordinates $x \in K^{n\geqslant 1}$, $y \in K$, $a \in K^{m\geqslant 1}$, $b \in K$ under split-diffeomorphisms $(x,y,a,b) \,\longmapsto\, \big( f(x,y),\,g(x,y),\,\varphi(a,b),\,\psi(a,b) \big)$. Two Levi forms exist, and have the same rank $r \leqslant \min (n,m)$. If $\mathcal{M}$ is $k$-nondegenerate with respect to parameters and $l$-nondegenerate with respect to variables, $\mbox{Aut}(\mathcal{M})$ is a local Lie group of dimension: $\dim\, \mbox{Aut} (\mathcal{M}) \,\,\leqslant\,\, {\textstyle{\binom{n+1+2k+2l}{2k+2l}}}\,\, \min\, \big\{ (n+1),\, (m+1) \big\}.$ Mainly, our goal is to set up foundational material addressed to CR geometers. We focus on $n = m = 2$, assuming $r = 1$. In coordinates $(x,y,z, a,b,c)$, a local equation is: $z \,=\, c + xa + \beta\,xxb + \underline{\beta}\,yaa + c\,{\rm O}_{x,y,a,b}(2) + {\rm O}_{x,y,a,b,c}(4),$ with $\beta$ and $\underline{\beta}$ representing the two $2$-nondegeneracy invariants at $0$. The associated para-CR PDE system: $z_y \,=\, F\big(x,y,z,z_x,z_{xx}\big) \ \ \ \ \ \ \ \ \ \ \ \ \ \& \ \ \ \ \ \ \ \ \ \ \ \ \ z_{xxx} \,=\, H\big(x,y,z,z_x,z_{xx}\big),$ satisfies $F_{z_{xx}} \equiv 0$ from Levi degeneracy.
We show in details that the hypothesis of $2$-nondegeneracy with respect to variables is equivalent to $F_{z_x z_x} \neq 0$. This gives CR-geometric meaning to the first two para-CR relative differential invariants encountered independently in arxiv.org/abs/2003.08166/.


Title: Critical metrics of the volume functional on three-dimensional manifolds
Authors: Huiya He
Subjects: Differential Geometry (math.DG)

In this paper, we prove the three-dimensional $CPE$ conjecture with non-negative Ricci curvature. Moreover, we show that a three-dimensional compact, oriented, connected Miao-Tam critical metric with smooth boundary, non-negative Ricci curvature and non-negative potential function is isometric to a geodesic ball in a simply connected space form $\mathbb{R}^3$ or $\mathbb{S}^3$. Finally, we establish a classification result on three-dimensional static space with non-negative Ricci curvature.


Title: Locally conformally balanced metrics on almost abelian Lie algebras
Subjects: Differential Geometry (math.DG)

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperk\"ahler structures.

### Cross-lists for Fri, 15 Jan 21

  arXiv:2101.05527 (cross-list from math.AP) [pdf, ps, other]
Title: Lojasiewicz inequalities for almost harmonic maps near simple bubble trees
Authors: Melanie Rupflin
Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)

We prove Lojasiewicz inequalities for the harmonic map energy for maps from surfaces of positive genus into general target manifolds which are close to simple bubble trees. Our results are not restricted to particular targets but apply whenever the underlying bubble is not branched, or more generally attached at a point which is not a branch point, and has integrable Jacobi fields.

  arXiv:2101.05643 (cross-list from math.AG) [pdf, ps, other]
Title: Some observations on the dimension of Fano K-moduli
Subjects: Algebraic Geometry (math.AG); Differential Geometry (math.DG)

In this short note we show the unboundedness of the dimension of the K-moduli space of $n$-dimensional Fano varieties, and that the dimension of the stack can also be unbounded while, simultaneously, the dimension of the corresponding coarse space remains bounded.

### Replacements for Fri, 15 Jan 21

  arXiv:1909.06347 (replaced) [pdf, ps, other]
Title: Steiner's formula and a variational proof of the isoperimetric inequality
Comments: Comments Welcome! v3 contains a new proof of one of the main lemmas, and a correspondingly simplified proof of the main result. The proof of the characterization of equality in this version also addresses a gap in this part of the proof in v1 and v2
Subjects: Differential Geometry (math.DG); Classical Analysis and ODEs (math.CA)
  arXiv:1910.06627 (replaced) [pdf, ps, other]
Title: Anosov representations with Lipschitz limit set
Comments: several changes in the introduction, minor corrections elsewhere
Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS); Group Theory (math.GR)
  arXiv:2006.01376 (replaced) [pdf, ps, other]
Title: Derived Differential Geometry
Subjects: Differential Geometry (math.DG); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Category Theory (math.CT)
  arXiv:2012.09127 (replaced) [pdf, other]
Title: Boundary conditions for scalar curvature
Comments: some material rearranged, presentation improved
Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)
  arXiv:1907.06150 (replaced) [pdf, ps, other]
Title: A KK-theoretic perspective on deformed Dirac operators
Subjects: K-Theory and Homology (math.KT); Differential Geometry (math.DG)
  arXiv:2003.03885 (replaced) [pdf, ps, other]
Title: Extending Nirenberg-Spencer's question on holomorphic embeddings to families of holomorphic embeddings
Authors: Jun-Muk Hwang
Comments: 24 pages. The result in Section 4 has been replaced by a weaker statement, because the argument in the first version had a gap, which was pointed out by Yong Hu. To appear in Duke Math. J
Subjects: Complex Variables (math.CV); Algebraic Geometry (math.AG); Differential Geometry (math.DG)
  arXiv:2003.06811 (replaced) [pdf, ps, other]
Title: Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations
Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG); Spectral Theory (math.SP)
  arXiv:2011.10030 (replaced) [pdf, ps, other]
Title: Differential forms on orbifolds with corners
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