A priori bounds and existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with log-convex measure

Zhengmao Chen; Zhengmao Chen

doi:10.3934/era.2023042

Electronic Research Archive

2023, Volume 31, Issue 2: 840-859. doi: 10.3934/era.2023042

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Research article

A priori bounds and existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with log-convex measure

Zhengmao Chen ^,

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

Academic Editor: Alain Miranville

Received: 31 August 2022 Revised: 20 November 2022 Accepted: 25 November 2022 Published: 01 December 2022

In the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.
- Codazzi tensor,
- log-convex measures,
- $ L_p $ Aleksandrov problem,
- the continuous method,
- a priori bounds
Citation: Zhengmao Chen. A priori bounds and existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with log-convex measure[J]. Electronic Research Archive, 2023, 31(2): 840-859. doi: 10.3934/era.2023042

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Abstract

In the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.

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