Curvature estimates for constant mean curvature surfaces

Almost Kähler manifolds of constant antiholomorphic sectional curvature

It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.

Kahler surfaces with quasi-constant holomorphic curvature

The aim of this paper is to describe Kahler surfaces with quasi-constant holomorphic curvature

Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form

We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of $M$.

Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is $0$ or $1$ and that the surface is ruled. If the generating curve is a Lorentzian circle, we show that the only possibility is that the axis is spacelike and the center of the circle lies in the axis.

Metrics of constant negative scalar-Weyl curvature

Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, t\in\mathbb{R}$. In particular, there are no topological obstructions for metrics with $\varepsilon$-pinched Weyl curvature and negative scalar curvature.

Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces

In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.

ParaSasakian manifolds with a constant paraholomorphic section curvature

In this paper paraSasakian manifolds with a constant paraholomorphic section curvature are considered.

Constant scalar curvature metrics on toric surfaces

This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/

Spherically Symmetric Finsler Metrics With Constant Ricci And Flag Curvature

Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.

Compactness of Constant Mean Curvature Surfaces in Three Manifold with Positive Ricci Curvature

In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.

Kropina spaces of constant curvature II (long version)

In this paper conditions for a Kropina structure to be of constant flag curvature are obtained.

On gradient Ricci solitons with constant scalar curvature

We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.

New classes of projectively related Finsler metrics of constant flag curvature

We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler metrics that have constant flag curvature.

Hermitian manifolds of pointwise constant antiholomorphic sectional curvatures

In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.

Entire bounded constant mean curvature Killing graphs

We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.

Biconservative hypersurfaces with constant scalar curvature in space forms

Biconservative hypersurfaces are hypersurfaces which have conservative stress-energy tensor with respect to the bienergy, containing all minimal and constant mean curvature hypersurfaces. The purpose of this paper is to study biconservative hypersurfaces $M^n$ with constant scalar curvature in a space form $N^{n+1}(c)$. We prove that every biconservative hypersurface with constant scalar curvature in $N^4(c)$ has constant mean curvature. Moreover, we prove that any biconservative hypersurface with constant scalar curvature in $N^5(c)$ is ether an open part of a certain rotational hypersurface …

Constant Curvature Conditions For Generalized Kropina Spaces

The classification of Finsler spaces of constant curvature is an interesting and important topic of research in differential geometry. In this paper we obtain necessary and sufficient conditions for generalized Kropina space to be of constant flag curvature.

An end-to-end construction for compact constant mean curvature surfaces

We explain how the current knowledge on the set of complete noncompact constant mean curvature surfaces can be exploited to produce new examples of compact constant mean curvature surfaces of genus greater than or equal to 3.

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

Hypersurfaces of Constant Curvature in Hyperbolic Space I

We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.

A note on starshaped hypersurfaces with almost constant mean curvature in space forms

We show that closed starshaped hypersurfaces of space forms with almost constant mean curvature or almost constant higher order mean curvature are closed to geodesic spheres.

Constant Mean Curvature Trinoids

This paper constructs a family of constant mean curvature immersions of the thrice-punctured Riemann sphere into Euclidean 3-space with asymptotically Delaunay ends via loop group methods.

Embedded minimal and constant mean curvature annulus touching spheres

We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.

常标量曲率与常边界平均曲率方程的爆破现象（继 Chen 和 Wu 之后）

Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation (after Chen and Wu)

针对常标量曲率与常边界平均曲率方程，本文将低能量解集失去紧性的光滑反例维数阈值由 Chen 和 Wu 的不小于62维改进至不小于35维。

Projective spherically symmetric Finsler metrics with constant flag curvature in R^n

We investigate projective spherically symmetric Finsler metrics with constant flag curvature in $R^n$ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk are found to have constant negative flag curvature.

具有常长度平均曲率向量的闭子流形的对数Sobolev不等式

A logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector

该研究在微分几何领域证明了非负截面曲率流形中具常长度平均曲率向量的闭子流形满足对数Sobolev不等式，丰富了子流形几何的分析工具。

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature $S$ of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of $S$. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli and Setti \cite{PRS}.

Construction of constant mean curvature n-noids using the DPW method

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

Geodesics in Randers spaces of constant curvature

Geodesics in Randers spaces of constant curvature are classified.

On stable hypersurfaces with constant mean curvature in Euclidean spaces

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.

Kahler manifolds with quasi-constant holomorphic curvature

The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.

Entire unbounded constant mean curvature Killing graphs

In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.

Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces

We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.

Umbilicity of constant mean curvature hypersurfaces into space forms

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.

Constant mean curvature surfaces of Delaunay type along a closed geodesic

In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.

Helicoidal surfaces with constant anisotropic mean curvature

We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature surfaces can be obtained by quadratures.

Constant mean curvature hypersurfaces condensing along a submanifold

Given any nondegenerate k-dimensional minimal submanifold K of codimension greater than 1, we prove the existence of families of constant mean curvature submanifolds, with mean curvature varying from one member of the family to another, which `condense' to K. In particular, our result proves the existence of constant mean curvature hypersurfaces with nontrivial topology in any Riemannian manifold.

Discrete constant mean curvature surfaces via conserved quantities

This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way.

Constant mean curvature n-noids in hyperbolic space

Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in hyperbolic space.

On a constant curvature statistical manifold

We will show that a statistical manifold $(M, g, \nabla)$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $\nabla$ is projectively flat and the curvatures satisfies $R=R^*$, where $R^*$ is the curvature of the dual connection $\nabla^*$. Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $-1$ statistical structures and vice versa.

Complete Constant Mean Curvature surfaces in homogeneous spaces

In this paper we classify complete surfaces of constant mean curvature whose Gaussian curvature does not change sign in a simply connected homogeneous manifold with a 4-dimensional isometry group.

Constant Q-curvature metrics on conic 4-manifolds

We consider the constant Q-curvature metric problem in the given conformal class on conic 4-manifolds and study related differential equations.

Hypersurfaces with constant anisotropic mean curvatures

We apply the evolution method to present a new proof of the Alexandrov type theorem for constant anisotropic mean curvature hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$.

Constant Mean Curvature Surfaces in $\mathbb{M}^2(c)\times\mathbb{R}$ and Finite Total Curvature

We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When $n=2$, our results are similar to those obtained in \cite{bds} for surfaces with constant mean curvature in space forms.

A Finsler metric of constant Gauss curvature $K = 1$ on 2-sphere

We construct a concrete example of constant Gauss curvature $K = 1$ on the 2-sphere having all geodesics closed and of same length.

Nearly Kähler manifolds of constant antiholomorphic sectional curvature

A classification theorem for nearly Kähler manifolds of constant antiholomorphic sectional curvature is proved.

Stable hypersurfaces with constant scalar curvature in Euclidean spaces

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int_M|A|^3<+\infty$).

Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

Some classifications of biharmonic hypersurfaces with constant scalar curvature

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic h…

Constant mean curvature Isometric Immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and related results

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3-$dimensional homogenous manifolds $\mathbb{E}(κ, τ)$ and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \time…

Uniqueness of the Foliation of Constant Mean Curvature Spheres in Asymptotically Flat 3-Manifolds

In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.

Stable hypersurfaces with constant higher order mean curvature

We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for CMC capillary hypersurfaces.

Hypersurfaces with constant sectional curvature of $\Sf^n\times \R$ and $\Hy^n\times \R$

We classify the hypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$ with constant sectional curvature and dimension $n\geq 3$.

A Hopf theorem for non-constant mean curvature and a conjecture of A.D. Alexandrov

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in R3.