Related papers: Rigidity of Lipschitz map using harmonic map heat …

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It…

In this paper, we give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain four dimensional fiber bundles by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau. As a corollary…

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower…

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems,…

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to…

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy…

Given any continuous, lower bounded and $\kappa$-convex function $V$ on a metric measure space $(X,d,m)$ which is infinitesimally Hilbertian and satisfies some synthetic lower bound for the Ricci curvature in the sense of…

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form…

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate…

In this work, we study a gap phenomenon in locally conformally flat Riemannian manifolds with non-negative Ricci curvature. We construct complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve.…

We study harmonic map heat flow along ancient super Ricci flow, and derive several Liouville theorems with controlled growth from Perelman's reduced geometric viewpoint. For non-positively curved target spaces, our growth condition is…

The purpose of this article is to study Lipschitz CR mappings from an $h$-extendible (or semi-regular) hypersurface in $\mbb C^n$. Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A…

For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $\Sigma M$, and any compact Riemannian manifold $N$, we show an $\epsilon$-regularity theorem for weakly Dirac-harmonic maps . As a…

We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently…

We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and…

In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be…

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial $\mu$-entropy. Our basic principle tells that once correctly renormalized,…

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number…

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry…

We study geometric and topological properties of the image of a smooth submanifold of $\mathbb{R}^{n}$ under a bi-Lipschitz map to $\mathbb{R}^{m}$. In particular, we characterize how the dimension, diameter, volume, and reach of the…