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Softcover ISBN:  9781470435240 
Product Code:  MEMO/259/1244 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
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Product Code:  MEMO/259/1244.E 
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AMS Member Price:  $48.60 
Softcover ISBN:  9781470435240 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 259; 2019; 83 ppMSC: Primary 35
In this paper the authors first obtain a constant rank theorem for the second fundamental form of the spacetime level sets of a spacetime quasiconcave solution of the heat equation. Utilizing this constant rank theorem, they obtain some strictly convexity results of the spatial and spacetime level sets of the spacetime quasiconcave solution of the heat equation in a convex ring. To explain their ideas and for completeness, the authors also review the constant rank theorem technique for the spacetime Hessian of spacetime convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

Table of Contents

Chapters

1. Introduction

2. Basic definitions and the Constant Rank Theorem technique

3. A microscopic spacetime Convexity Principle for spacetime level sets

4. The Strict Convexity of Spacetime Level Sets

5. Appendix: the proof in dimension $n=2$


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

Chapters

1. Introduction

2. Basic definitions and the Constant Rank Theorem technique

3. A microscopic spacetime Convexity Principle for spacetime level sets

4. The Strict Convexity of Spacetime Level Sets

5. Appendix: the proof in dimension $n=2$