"In this Memoir 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 spacetime Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function"--
Chen, Ma, and Salani begin by obtaining a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasi-concave solution of the heat equation. Using the constant rank theorem, they obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasi-concave solution of the heat equation in a convex ring. In order to explain their ideas and for completeness, they also review the constant rank theorem technique for the space-time Hessian of space-time convex solutions of the heat equation and for the second fundamental form of the convex level sets for harmonic function. Annotation ©2019 Ringgold, Inc., Portland, OR (protoview.com)
Introduction
Basic definitions and the constant rank theorem technique
A microscopic space-time convexity principle for space-time level sets
The strict convexity of space-time level sets
Appendix: the proof in dimension $n=2$
Bibliography.
Chuanqiang Chen, Zhejiang University of Technology, Hangzhou, China.
Xinan Ma, University of Science and Technology of China, Hefei, China.
Paolo Salani, Universita di Firenze, Italy.
