arXiv subjects
Publications and source records attributed to Jiangcheng You.
We prove that a contractible $3$-manifold admitting a complete Riemannian metric of nonnegative scalar curvature is diffeomorphic to $\R^3$. We also prove that, if the interior $M_\gamma$ of a compact handlebody of genus $\gamma$ admits such a metric, then $\gamma\leq1$. This answers two open questions posed by Wang and Gromov, respectively.
We study a Brown--York type mass for closed hypersurfaces in four-dimensional asymptotically flat manifolds. The reference mean curvature is defined intrinsically as the trace of the positive solution of the contracted Gauss equation. For large uniformly convex hypersurfaces with controlled scale, we derive an expansion consisting of a boundary term converging to the ADM mass and a shape-dependent correction. For the four-dimensional analogue of the nearly round surfaces of Shi--Wang--Wu, this correction vanishes under a natural decay compatibility condition.
Let $M^n_\kappa$ be the simply connected space form of dimension $n\ge2$ and constant sectional curvature $\kappa\in\{-1,1\}$. For every bounded connected smooth domain $\Omega\subset M^n_\kappa$, assume in the case $\kappa=1$ that $\Omega$ is contained in an open hemisphere, and let $B_\Omega$ be a geodesic ball with $|B_\Omega|=|\Omega|$. We prove $$ \sum_{j=1}^n \frac1{\mu_j(\Omega)}\ge \frac{n}{\mu_1(B_\Omega)}, $$ where $\mu_j(\Omega)$ are the positive Neumann eigenvalues of $\Omega$. Equality holds if and only if $\Omega$ is a geodesic ball. This proves a conjecture proposed by Xia and Wang [Math. Ann. 385, 2023, 863-879].
We study refined asymptotics of scalar-flat ALE four-manifolds in the Tian--Viaclovsky setting, namely for self-dual or anti-self-dual metrics and for metrics with harmonic curvature. Starting from the ALE coordinates obtained by Tian--Viaclovsky, we construct preferred coordinates at infinity and identify the homogeneous $|x|^{-2}$ term in the metric expansion. This term splits canonically into a scalar part determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. As an application, we consider scalar-flat K\"ahler ALE metrics on minimal resolutions $\pi:X\to\mathbb C^2/\Gamma$ of quotient surface singularities. In this case, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.
We prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture. Let $g$ be a smooth complete metric on $\R^3$ with non-negative scalar curvature. If $$ |g-g_{\Euc}|=o(r^{-1}),\qquad r=|x|\to\infty, $$ then $(\R^3,g)$ is isometric to Euclidean space.