arXiv subjects
Publications and source records attributed to Tianyin Ren.
In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant $α=-1$ (or $+1$), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.
In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete $n$-manifold $M$ of Ricci curvature, $\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $π$, then $M$ is isometric to the unit sphere $S^n_1$. The main result in this paper is a quantitative maximal diameter rigidity: if $M$ satisfies that $\operatorname{Ric}_M\ge n-1$, $\operatorname{diam}(M)\approx π$, and the Riemannian universal cover of every metric ball in $M$ of a definite radius satisfies a Riefenberg condition, then $M$ is diffeomorphic and bi-Hölder close to $S^n_1$.