Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2023; 60(4): 1085-1100

Online first article July 7, 2023      Printed July 31, 2023

https://doi.org/10.4134/BKMS.b220531

Copyright © The Korean Mathematical Society.

Hypersurfaces with prescribed mean curvature in measure metric space

Zhengmao Chen

Guangxi Normal University

Abstract

For any given function $f$, we focus on the so-called prescribed mean curvature problem for the measure $e^{-f(|x|^2)}dx$ provided that $e^{-f(|x|^2)}\in L^1(\mathbb{R}^{n+1})$. More precisely, we prove that there exists a smooth hypersurface $M$ whose metric is $ds^2=d\rho^2+\rho^2d\xi^2$ and whose mean curvature function is \begin{equation*} \frac{1}{n}\frac{u^p}{\rho^\beta}e^{f(\rho^2)}\psi(\xi) \end{equation*} for any given real constants $p$, $\beta$ and functions $f$ and $\psi$ where $u$ and $\rho$ are the support function and radial function of $M$, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere $\mathbb{S}^{n}$, \begin{equation*} \sum\limits_{i,j}(\delta_{ij}-\frac{\rho_i\rho_j}{\rho^2+|\nabla\rho|^2})(-\rho_{ji} +\frac{2}{\rho}\rho_j\rho_i +\rho\delta_{ji})=\psi\frac{\rho^{2p+2-n-\beta} e^{f(\rho^2)}}{(\rho^2+|\nabla \rho|^2)^{\frac{p}{2}}} \end{equation*} under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)=\frac{t}{2}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

Keywords: Measure metric space, mean curvature, quasilinear equation, the continuous method

MSC numbers: Primary 53A10, 53C42, 35J15

Supported by: The work was supported by The Science and Technology Project of Guangxi (Guike AD21220114), China Postdoctoral Science Foundation (Grant: No.2021M690773) and Key Laboratory of Mathematical and Statistical Model (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region.