# Analysis Seminar: Asaf Shachar (HUJI) — Non-Euclidean elasticity: Embedding manifolds with minimal distortion

Location: Ross 70A and Zoom
https://huji.zoom.us/j/88693310302?pwd=cjZDOWhhMmJSaFVrdHhMZ0JlOVRHUT09

Title: Non-Euclidean elasticity: Embedding manifolds with minimal distortion

Abstract: Given two dimensional Riemannian manifolds $M,N$, I will present a sharp lower bound on the elastic energy (distortion) of embeddings $f:M \to N$, in terms of the areas' discrepancy of $M,N$.
The minimizing maps attaining this bound go through a phase transition when the ratio of areas is $1/4$: The homotheties are the unique energy minimizers when the ratio $\frac{V_N}{V_M} \ge 1/4$, and they cease being minimizers when $\frac{V_N}{V_M}$ gets below $1/4$.
I will describe explicit minimizers in the non-trivial regime $\frac{V_N}{V_M} < 1/4$ when $M,N$ are disks, and give a proof sketch of the lower bound. If time permits, I will discuss the stability of minimizers.

## Date:

Wed, 21/04/2021 - 12:00 to 13:00
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9772149920463562, "perplexity": 986.5338364802583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00187.warc.gz"}