5,253 research outputs found
Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4-manifolds
The goal of this article is to establish estimates involving the Yamabe
minimal volume, mixed minimal volume and some topological invariants on compact
4-manifolds. In addition, we provide topological sphere theorems for compact
submanifolds of spheres and Euclidean spaces, provided that the full norm of
the second fundamental form is suitably bounded.Comment: To appear in Mathematische Nachrichte
4-dimensional compact manifolds with nonnegative biorthogonal curvature
The goal of this article is to study the pinching problem proposed by S.-T.
Yau in 1990 replacing sectional curvature by one weaker condition on
biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented
Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we
obtain a partial answer to Yau Conjecture on pinching theorem for 4-dimensional
compact manifolds.Comment: To appear in the Michigan Mathematical Journa
Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary
We provide a general B\"ochner type formula which enables us to prove some
rigidity results for -static spaces. In particular, we show that an
-dimensional positive static triple with connected boundary and positive
scalar curvature must be isometric to the standard hemisphere, provided that
the metric has zero radial Weyl curvature and satisfies a suitable pinching
condition. Moreover, we classify -static spaces with non-negative sectional
curvature.Comment: Fixed typo
Bach-flat noncompact steady quasi-Einstein manifolds
The goal of this article is to study the geometry of Bach-flat noncompact
steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady
quasi-Einstein manifold with positive Ricci curvature such that
its potential function has at least one critical point must be a warped product
with Einstein fiber. In addition, the fiber has constant curvature if Comment: To appear in Archiv der Mathemati
A note on the uniqueness of quasi-Einstein metrics on
The aim of this note is to give an explicit description of quasi-Einstein
metrics on We shall construct two examples of
quasi-Einstein metrics on this manifold and then we shall prove the uniqueness
of these examples. Finally, we shall describe the closed relation between
quasi-Einstein metrics and static metrics in the quoted space.Comment: Updated version. All comments are welcome
Four-manifolds with positive curvature
In this note we prove that a four-dimensional compact oriented
half-confor\-mally flat Riemannian manifold is topologically
or provided that the sectional
curvatures all lie in the interval In addition,
we use the notion of biorthogonal (sectional) curvature to obtain a pinching
condition which guarantees that a four-dimensional compact manifold is
homeomorphic to a connected sum of copies of the complex projective plane or
the -sphere.Comment: Revised versio
Estimates for Minimal Volume and Minimal Curvature on 4-dimensional compact manifolds
In a remarkable article published in 1982, M. Gromov introduced the concept
of minimal volume, namely, the minimal volume of a manifold is defined to
be the greatest lower bound of the total volumes of with respect to
complete Riemannian metrics whose sectional curvature is bounded above in
absolute value by 1. While the minimal curvature, introduced by G. Yun in 1996,
is the smallest pinching of the sectional curvature among metrics of volume 1.
The goal of this article is to provide estimates to minimal volume and minimal
curvature on 4-dimensional compact manifolds involving some differential and
topological invariants. Among these ones, we get some sharp estimates for
minimal curvature.Comment: revised versio
Volume functional of compact -manifolds with a prescribed boundary metric
We prove that a critical metric of the volume functional on a -dimensional
compact manifold with boundary satisfying a second-order vanishing condition on
the Weyl tensor must be isometric to a geodesic ball in a simply connected
space form , or Moreover, we
provide an integral curvature estimate involving the Yamabe constant for
critical metrics of the volume functional, which allows us to get a rigidity
result for such critical metrics.Comment: To appear in The Journal of Geometric Analysi
Geometric Inequalities for Critical Metrics of the Volume Functional
The goal of this article is to investigate the geometry of critical metrics
of the volume functional on an -dimensional compact manifold with (possibly
disconnected) boundary. We establish sharp estimates to the mean curvature and
area of the boundary components of critical metrics of the volume functional on
a compact manifold. In addition, localized version estimates to the mean
curvature and area of the boundary of critical metrics are also obtained.Comment: Fixed typo
Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary
The purpose of this article is to investigate Bach-flat critical metrics of
the volume functional on a compact manifold with boundary
Here, we prove that a Bach-flat critical metric of the volume functional on a
simply connected 4-dimensional manifold with boundary isometric to a standard
sphere must be isometric to a geodesic ball in a simply connected space form
or . Moreover, we show that in
dimension three the result even is true replacing the Bach-flat condition by
the weaker assumption that has divergence-free Bach tensor.Comment: To appear in The Journal of Geometric Analysi
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