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Yuxin Ge

Publications and source records attributed to Yuxin Ge.

24 records · Page 2Linked to original sources
arXiv2013

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_Σ\s_4 d μ\ge \ds\vs C_{n-1}^4ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 12 + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array}\] provided that $Σ$ is a horospherical convex hypersurface. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$.

math.DG↗
arXiv2012

Locally nearly spherical surfaces are almost-positively $c$-curved

The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C^2$ norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant curvature case, keeping track of an estimate on the closeness curvature condition.

math.DG↗
arXiv2011

A new conformal invariant on 3-dimensional manifolds

By improving the analysis developed in the study of $\s_k$-Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if $(M^3, g)$ is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, then \[\int_M |Ric-\frac{\bar R} 3 g|^2 dv (g)\le 9\int_M |Ric-\frac{R} 3 g|^2dv(g), \] where $\bar R=vol (g)^{-1} \int_M R dv(g)$ is the average of the scalar curvature $R$ of $g$. Equality holds if and only if $(M^3,g)$ is a space form. We in fact study the following new conformal invariant \[\ds \widetilde Y([g_0]):=\sup_{g\in {\cal C}_1([g_0])}\frac {\ds vol(g)\int_M \s_2(g) dv(g)} {\ds (\int_M \s_1(g) dv(g))^2}, \] where ${\cal C}_1([g_0]):=\{g=e^{-2u}g_0\,|\, R>0\}$ and prove that $\widetilde Y([g_0])\le 1/3$, which implies the above inequality.

math.DG↗
arXiv2004

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem $Δu+ λu + u^p = 0$ with Dirichet boundary conditions, in a bounded smooth domain $Ω\subset \R^N$ $(N\geq 4)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$ and the parameter $λ\in\R$ is small enough. As $p\to \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino, Dolbeault and Musso \cite{DDM} when $Ω$ is a ball and the solutions are radially symmetric.

math.AP↗