Related papers: A rigidity theorem for holomorphic disks in Teichm…

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric…

We provide a complete classification of Teichm\"uller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichm\"uller curves intersect the boundary of the…

The rigidity theorem for homotopy invariant presheaves with Witt-transfers on the category of smooth affine varieties over a field $k$ with characteristic not equal to 2 is proved. Namely for such a presheaf $\mathcal F$ the isomorphism…

Let $\varphi :\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_{\varphi}:\mathbb{D}\to T(\mathbb{D})$ the Teichm\"uller disk in the universal Teichm\"uller space $T(\mathbb{D})$. For a…

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…

I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi}{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere can be rigid under smooth deformations that increase…

An inequality on torsional rigidity is established. For tangential polygons this inequality is stronger than an inequality of Polya and Szego for convex domains. (A survey of related work, not in the journal submission, is presented in the…

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds with universal cover a bounded domain is a finite set.

In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which…

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a…

Let $f:X\to Y$ be a morphism of complex manifolds. Suppose that $X$ is a K\"ahler manifold. Let $(\mathcal{T},\mathcal{S})$ be a regular polarized pure twistor $\mathcal{D}$-module of weight $w$ on $X$ whose support is proper over $Y$. We…

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify…

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.

Perez proved some $L^2$ inequalities for closed convex hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, more generally, for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this…

Teichm\"uller curves play an important role in the study of dynamics in polygonal billiards. In this article, we provide a criterion similar to the original M\"oller's criterion, to detect whether a complex curve, embedded in the moduli…

In this paper, we establish a Liouville type rigidity result for a class of asymptotically hyperbolic non-compact Einstein metrics defined on manifolds of dimension $d\ge 5$ extending the earlier result in dimension $d=4$.

Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of…

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem…

We obtain a generalization of the Burns-Krantz rigidity theorem for holomorphic self-mappings of the unit disk in the spirit of the classical Schwarz-Pick Lemma and its continuous version due to L.Harris via the generation theory for…

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min-Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.