arXiv subjects
Publications and source records attributed to Zhangkai Huang.
We prove an integral type Gauss-Green formula on non-collapsed RCD spaces using the strong locality of the Laplacian and an eigenfunction approximation method. As applications, we generalize Colding's monotonicity formulas and prove an asymptotic formula linking the mean curvature of a hypersurface at a given point to the volume of small balls centered at that point.
In this paper, we study a family of $n$-dimensional Riemannian manifolds with boundary having lower bounds on the Ricci curvatures of interior and boundary and on the second fundamental form of boundary. A sequence of manifolds in this family is said to be inradius collapsed if their inradii tend to zero. We prove that the limit space $C_0$ of boundaries of inradius collapsed manifolds admits an isometric involution $f$, and that the limit of the manifolds themselves is isometric to the quotient space $C_0/f$. As an application, we show that the number of boundary components of inradius collapsed manifolds is at most two. Moreover, we prove that the limit space has a lower Ricci curvature bound and an upper dimension bound in a synthetic sense if in addition their boundaries are non-collapsed.
In this paper, we study harmonic RCD$(K,N)$ spaces as the counterpart of harmonic Riemannian manifolds with Ricci curvature bounded from below. We prove that a compact RCD$(K,N)$ space is isometric to a smooth closed Riemannian manifold if it satisfies either of the following harmonicity conditions:(1) the heat kernel $ρ(x,y,t)$ depends only on the variable $t$ and the distance between points $x$ and $y$; (2) the volume of the intersection of two geodesic balls depends only on their radii and the distance between their centers.
Given an RCD$(K,N)$ space $({X},\mathsf{d},\mathfrak{m})$, one can use its heat kernel $ρ$ to map it into the $L^2$ space by a locally Lipschitz map $Φ_t(x):=ρ(x,\cdot,t)$. The space $(X,\mathsf{d},\mathfrak{m})$ is said to be an isometrically heat kernel immersing space, if each $Φ_t$ is an isometric immersion {}{after a normalization}. A main result states that any compact isometrically heat kernel immersing RCD$(K,N)$ space is isometric to an unweighted closed smooth Riemannian manifold. This is justified by a more general result: if a compact non-collapsed RCD$(K, N)$ space has an isometrically immersing eigenmap, then the space is isometric to an unweighted closed Riemannian manifold, which greatly improves a regularity result in \cite{H21} by Honda. As an application of these results, we give a $C^\infty$-compactness theorem for a certain class of Riemannian manifolds with a curvature-dimension-diameter bound and an isometrically immersing eigenmap.