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

In this paper we develop the compactness theorem for $\lambda$-surface in $\mathbb R^3$ with uniform $\lambda$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for…

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of…

The density property for a Stein manifold X implies that the group of holomorphic diffeomorphisms of X is infinite-dimensional and, in a certain well-defined sense, as large as possible. We prove that if G is a complex semisimple Lie group…

We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq…

We study symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our main result can be seen as a generalization to open Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on the Nearby…

The main theorem of the paper provides an existence criterion of holomorphic discs for higher $A_\infty$ operations. The key step is to show that if a minimal disc in a K\"ahler manifold with boundary in a sequence of Lagrangian…

The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application…

A theorem of Kuranishi tells us that the moduli space of complex structures on any smooth compact manifold is always locally a finite-dimensional space. Globally, however, this is simply not true; we display examples in which the moduli…

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for…

Our main result is the $\mathcal{C}^0$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic discs in cotangent bundles with boundary on the zero section,…

Haslhofer and M\"uller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly…

We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the…

In this article, we revisit classical length identities enjoyed by simple closed curves on hyperbolic surfaces. We state and prove the rigidity of such identities over Teichm\"uller spaces. Due to this rigidity, certain collections of…

A Teichm\"uller space $Teich$ is a quotient of the space of all complex structures on a given manifold $M$ by the connected components of the group of diffeomorphisms. The mapping class group $\Gamma$ of $M$ is the group of connected…

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is…

Following ideas of Choe and Fernandez-do Carmo, we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for…

In this paper we study the rigidity problem for sub-static systems with possibly non-empty boundary. First, we get local and global splitting theorems by assuming the existence of suitable compact minimal hypersurfaces, complementing recent…

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…

Motivated by recent papers \cite{For-Rong 2021} and \cite{Ng-Rong 2024} we prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool in the proofs…

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $\pi_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichm\"uller…