Abstract
We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hy-perplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and hipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincare inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.
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Dedicated to Carlos Kenig on his 65th birthday with great affection
Supported in part by NSF (Grant No. DMS 1500771), a Simons Fellowship, and the Simons Foundation (Grant No. 601948 DJ)
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Jerison, D. The Two Hyperplane Conjecture. Acta. Math. Sin.-English Ser. 35, 728–748 (2019). https://doi.org/10.1007/s10114-019-8241-8
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DOI: https://doi.org/10.1007/s10114-019-8241-8