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# memo_has_moved_text();On Space-Time Quasiconcave Solutions of the Heat Equation

Chuanqiang Chen, Xinan Ma and Paolo Salani

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 259, Number 1244
ISBNs: 978-1-4704-3524-0 (print); 978-1-4704-5243-8 (online)
DOI: https://doi.org/10.1090/memo/1244
Published electronically: April 12, 2019
Keywords:Heat equation, quasiconcavity, space-time level set, constant rank theorem, space-time quasiconcave solution

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Chapters

• Chapter 1. Introduction
• Chapter 2. Basic definitions and the Constant Rank Theorem technique
• Chapter 3. A microscopic space-time Convexity Principle for space-time level sets
• Chapter 4. The Strict Convexity of Space-time Level Sets
• Chapter 5. Appendix: the proof in dimension $n=2$

### Abstract

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.