13,802 research outputs found
Equivariant spectral flow and a Lefschetz theorem on odd dimensional spin manifolds
A notion of equivariant spectral flows for families of self-dual elliptic
operators on Riemannian manifolds is purposed. As a consequence, a local
version of a Lefschetz fix point theorem is proved for Toeplitz operators on
odd-dimensional spin manifolds.Comment: 13 page
On a multi-particle Moser-Trudinger Inequality
We verify a conjecture of Gillet-Soul\'{e}. We prove that the determinant of
the Laplacian on a line bundle over is always bounded from
above. This can also be viewed as a multi-particle generalization of the
Moser-Trudinger Inequality. Furthermore, we conjecture that this functional
achieves its maximum at the canonical metric. We give some evidence for this
conjecture, as well as links to other fields of analysis
On a Conformal Gauss-Bonnet-Chern inequality for LCF manifolds and related topics
In this paper, we prove the following two results:
First, we study a class of conformally invariant operators and their
related conformally invariant curvatures on even-dimensional Riemannian
manifolds. When the manifold is locally conformally flat(LCF) and compact
without boundary, -curvature is naturally related to the integrand in the
classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature. For a class
of even-dimensional complete LCF manifolds with integrable % -curvature, we
establish a Gauss-Bonnet-Chern inequality.
Second, a finiteness theorem for certain classes of complete LCF four-fold
with integrable Pfaffian curvature is also proven. This is an extension of the
classical results of Cohn-Vossen and Huber in dimension two. It also can be
viewed as a fully non-linear analogue of results of Chang-Qing-Yang in
dimension four.Comment: 26 pages, 0 figur
On convergence to a football
We show that spheres of positive constant curvature with () conic
points converge to a sphere of positive constant curvature with two conic
points (or called an (American) football) in Gromov-Hausdorff topology when the
corresponding singular divisors converge to a critical divisor in the sense of
Troyanov.
We prove this convergence in two different ways. Geometrically, the
convergence follows from Luo-Tian's explicit description of conic spheres as
boundaries of convex polytopes in . Analytically, regarding the
conformal factors as the singular solutions to the corresponding PDE, we derive
the required a priori estimates and convergence result after proper
reparametrization.Comment: 19 page
Torsion type invariants of singularities
Inspired by the LG/CY correspondence, we study the local index theory of the
Schr\"odinger operator associated to a singularity defined on
by a quasi-homogeneous polynomial . Under some mild assumption on , we
show that the small time heat kernel expansion of the corresponding
Schr\"odinger operator exists and is a series of fractional powers of time .
Then we prove a local index formula which expresses the Milnor number of by
a Gaussian type integral. Furthermore, the heat kernel expansion provides
spectral invariants of . Especially, we define torsion type invariants
associated to a singularity. These spectral invariants provide a new direction
to study the singularity
On the Mean Curvature Flow for -Convex Hypersurfaces
We obtain estimates on both size and dimensions of the singular set at the
first blow-up time of the mean curvature flow of hypersurfaces whose initial
data is -convex.Comment: to appear in Houston Journal of Mathematic
Analytic Torsion and R-Torsion for Manifolds with Boundary
We prove a formula relating the analytic torsion and Reidemeister torsion on
manifolds with boundary in the general case when the metric is not necessarily
a product near the boundary. The product case has been established by W. Lu\"ck
and S. M. Vishik. We find that the extra term that comes in here in the
nonproduct case is the transgression of the Euler class in the even dimensional
case and a slightly more mysterious term involving the second fundamental form
of the boundary and the curvature tensor of the manifold in the odd dimensional
case.Comment: 24 pages. to appear in Asian J. Mat
On the geometric flows solving K\"ahlerian inverse equations
In this note, we extend our previous work on the inverse problem.
Inverse problem is a fully nonlinear geometric PDE on compact
K\"ahler manifolds. Given a proper geometric condition, we prove that a large
family of nonlinear geometric flows converges to the desired solution of the
given PDE.Comment: to appear in Pacific Journal of Mathematic
Volume bounds of conic 2-spheres
We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian
curvature bound. We also give the geometric models realizing the extremal
volume. In particular, when the curvature is bounded in absolute value by ,
we compute the minimal volume of a conic sphere in the sense of Gromov. In
order to apply the level set analysis and iso-perimetric inequality as in our
previous works, we develop some new analytical tools to treat regions with
vanishing curvature.Comment: 19 pages, 1 figur
Yamabe problem on conic 4-spheres
We discuss the constant problem for conic 4-spheres. Based on
earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a
necessary condition for the existence problem. In particular, when the
condition is sharp, we have the uniqueness result similar to that of Troyanov
in dimension 2. It indicates that the boundary of the moduli of all conic
4-spheres with constant metrics consists of conic spheres with 2
conic points and rotational symmetry.Comment: 20 pages, we makes some changes in the paper posted befor
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