Related papers: On the $\sigma_{k}$-Nirenberg problem

In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which…

We prove that if a closed Riemannian manifold $(M^n,g)$ has finite fundamental group and satisfies the curvature condition \begin{equation*} R_{1313} +R_{1414} +R_{2323} + R_{2424} > \tfrac{1}{2}\left(R_{1212} + R_{3434}\right)…

In this paper, we prove that, for any integer $n\ge 2,$ there exists an $\epsilon_{n} \ge 0$ so that if $M$ is an n-dimensional complete manifold with sectional curvature $ K_{M}\ge 1$ and if $M$ has conjugate radius bigger than…

We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\partial K$ are larger than or equal to…

In this paper, we investigate the noncompact prescribed Chern scalar curvature problem which reduces to solve a Kazdan-Warner type equation on noncompact non-K\"{a}hler manifolds. By introducing an analytic condition on noncompact…

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants \sigma(M) and \sigma(N) satisfy \sigma(N)\ge min (\sigma(M),\Lambda) for \Lambda>0. We…

We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that…

In this paper we give a pinching theorem of the Simon conjecture in the case s=3 and also give a new proof of the cases s=1 and s=2 by some Simons-type integral inequalities.

On the two-sphere $\Sigma$, we consider the problem of minimising among suitable immersions $f \,\colon \Sigma \rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient…

In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to $S^1 \times S^{n-1}$ for each $n \geq 2$ in the…

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - \kappa_i) = \big( (n - 1) \sigma \big)^n$ for $n = 4$ and has prescribed asymptotic boundary…

On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was…

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 use transference principle to show that whenever $s$ is suitably large depending on $k \geq 2$, every sufficiently large natural number $n$ satisfying some congruence conditions can be written in the form $n = p_1^k + \dots + p_s^k$,…

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $\lambda_p$'s are…

In this paper, we successfully set up a generalized sphere theorem for compact Riemannian manifolds with radial Ricci curvature bounded.

Is a sequence of Riemannian manifolds with positive scalar curvature, satisfying some conditions to keep the sequence reasonable, compact? What topology should one use for the convergence and what is the regularity of the limit space? In…

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…

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the…

This is the second of two papers that describe a compactness theorem for sequences of solutions of certain SL(2;C) analogs of the anti-self dual equations on oriented, 4-dimensional Riemannian manifolds. This paper proves theorems that…