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Title: Conformal Metrics on the Unit Ball with the Prescribed Mean Curvature
Keywords: conformal mean curvature flow, blow-up analysis, spectral decomposition, Morse theory
Issue Date: 17-Mar-2014
Citation: ZHANG HONG (2014-03-17). Conformal Metrics on the Unit Ball with the Prescribed Mean Curvature. ScholarBank@NUS Repository.
Abstract: This thesis focuses on the prescribed mean curvature problem on the unit ball in the Euclidean space with dimension three or higher. Such problem is well known and attracts a lot of attention. If the candidate $f$ for the prescribed mean curvature is sufficiently close to the mean curvature of the standard metric in the sup norm, then the existence of solution has been known for more than fifteen years. It is interesting to investigate how large that difference can be. This thesis partially achieves this goal using the mean curvature flow method. More precisely, we assume that the given candidate $f$ is a smooth positive Morse function which is non-degenerate in the sense that $|\nabla f|_{S^n}^2+(\Delta_{S^n}f)^2\neq0$ and $\mbox{max}_{S^n}f/\mbox{min}_{S^n}f<\delta_n$, where $\delta_n=2^{1/n}$, when $n=2$ and $\delta_n=2^{1/(n-1)}$, when $n\ge3$. We then show that $f$ can be realized as the mean curvature of some conformal metric provided the Morse index counting condition holds for $f$. This shows that the possible best difference in the sup norm may be the number $(\delta_n-1)/(\delta_n+1)$.
Appears in Collections:Ph.D Theses (Open)

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