Related papers: Einstein submanifolds with parallel mean curvature
Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question…
The main aim of this survey paper is to gather together some results concerning the Calabi type duality discovered by Hojoo Lee between certain families of (spacelike) graphs with constant mean curvature in Riemannian and Lorentzian…
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface $\Gamma$ in a Cartan-Hadamard manifold…
Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes…
Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe…
Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin$^c$ case the result of O.…
This survey paper investigates, from a purely geometric point of view, Daniel's isometric conjugation between minimal and constant mean curvature surfaces immersed in homogeneous Riemannian three-manifolds with isometry group of dimension…
In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in…
We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another…
The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article,…
We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures…
We prove that there are infinitely many pairs of homeomorphic non-diffeomorphic smooth 4-manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4-manifolds with…
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel…
An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of…
Uniqueness and non-existence results on complete constant mean curvature spacelike hypersurfaces lying between two spacelike slice in the Einstein-de Sitter spacetime are given. They are obtained from a Liouvielle-type theorem applied to a…
Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On…
We prove that under certain conditions on the mean curvature and on the Kaehler angles, a compact submanifold M of real dimension 2n, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, must be either a complex or a…
We study a variational problem on a smooth manifold with a decomposition of the tangent bundle into $k>2$ subbundles (distributions), namely, we consider the integrated sum of their mixed scalar curvatures as a functional of adapted…
An Einstein manifold in four dimensions has some configuration of $SU(2)_+$ Yang-Mills instantons and $SU(2)_-$ anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group…
We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous manifold with a 4-dimensional isometry group. The condition is expressed in terms of…