Related papers: Linear Quantitative Rigidity for Almost-CMC Surfac…

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4\pi/c$,…

In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.

In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds $M^3$ with nonnegative Ricci curvature and strictly convex boundary $\partial M$. Here we obtain a sharp upper bound for the length…

We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced.…

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant {\theta}-holomorphic sectional curvature for a number {\theta} in (0,{\pi}/2) then M is of…

An almost Fuchsian manifold is a hyperbolic 3-manifold of the type $S\times \mathbb{R}$ which admits a closed minimal surface (homeomorphic to $S$) with the maximum principal curvature $\lambda_0 <1$, while a weakly almost Fuchsian manifold…

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic…

We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as…

We show that the upper bounds for the $L^2$-norms of $L^1$-normalized quasimodes that we obtained in [9] are always sharp on any compact space form. This allows us to characterize compact manifolds of constant sectional curvature using the…

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 continue the analysis of random series associated to the multidimensional harmonic oscillator $-\Delta + |x|^2$ on $\mathbb{R}^d$ with d \geq 2$$. More precisely we obtain a necessary and sufficient condition to get the almost sure…

In this article, we construct complete embedded constant mean curvature surfaces in $\mb{R}^3$ with freely prescribed genus and any number of ends greater than or equal to four. Heuristically, the surfaces are obtained by resolving finitely…

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected…

This paper generalizes D. Burago and S. Ivanov's work on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal…

In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented $n$-dimensional ($n\geq6$) Riemannian manifold $(M,g)$ and prove the following results under the condition $\int_{M} \nabla R\cdot\nabla…

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change…

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals $\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g dV_g$, $t\in\mathbb{R}$, and…

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric…