Related papers: An exotic sphere with positive sectional curvature

Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we…

We give a new proof that the completion of the Weil-Petersson metric on Teichm\"uller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of…

The K\"ahler cone of a compact manifold carries a natural Riemannian metric, given by the intersection product of its cohomology ring. We write down the curvature tensor of this metric by embedding the K\"ahler cone in the space of…

Let $\Omega$ be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over $\hbox{Reg}(\Omega)$ is K\"ahler-Einstein.

Following P. M. H. Wilson's paper on sectional curvatures of Kahler moduli, we consider a natural Riemannian metric on a hypersurface f=1 in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to…

We prove for $n\in\{3,4,5\}$ that the connected sum of a closed aspherical $n$-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result…

Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain…

We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of…

We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G…

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit…

Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. By using Strebel differentials as a bridge, we construct a new class of cone spherical metrics on…

Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new…

We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with…

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such…

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero…

We investigate CMC-surfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMC-surface, we develop an…

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…

This paper is dedicated to scrutinizing the cosmology in massive gravity. A matter field of the dark sector is coupled to an effective composite metric while a standard matter field couples to the dynamical metric in the usual way. For this…