A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms

Zhengmao Chen; Zhengmao Chen

doi:10.3934/math.2023663

AIMS Mathematics

2023, Volume 8, Issue 6: 13134-13153. doi: 10.3934/math.2023663

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Research article

A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms

Zhengmao Chen ^,

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received: 10 February 2023 Revised: 17 March 2023 Accepted: 17 March 2023 Published: 03 April 2023
MSC : 35B45, 35J96, 53C42

In the present paper, we prove the a priori bounds and existence of smooth solutions to a Minkowski type problem for the log-concave measure $ e^{-f(|x|^2)}dx $ in warped product space forms with zero sectional curvature. Our proof is based on the method of continuity. The crucial factor of the analysis is the a priori bounds of an auxiliary Monge-Ampère equation on $ \mathbb{S}^n $. The main result of the present paper extends the Minkowski type problem of log-concave measures to the space forms and it may be an attempt to get some new analysis for the log-concave measures.
- log-concave measure,
- Minkowski problem,
- Monge-Ampère equation,
- the continuous method,
- warped product space forms
Citation: Zhengmao Chen. A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms[J]. AIMS Mathematics, 2023, 8(6): 13134-13153. doi: 10.3934/math.2023663

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Abstract

In the present paper, we prove the a priori bounds and existence of smooth solutions to a Minkowski type problem for the log-concave measure $ e^{-f(|x|^2)}dx $ in warped product space forms with zero sectional curvature. Our proof is based on the method of continuity. The crucial factor of the analysis is the a priori bounds of an auxiliary Monge-Ampère equation on $ \mathbb{S}^n $. The main result of the present paper extends the Minkowski type problem of log-concave measures to the space forms and it may be an attempt to get some new analysis for the log-concave measures.

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