Featured Researches

Differential Geometry

A Strong Maximum Principle for Weak Solutions of Quasi-Linear Elliptic Equations with Applications to Lorentzian and Riemannian Geometry

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for C 0 spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.

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Differential Geometry

A canonical way to deform a Lagrangian submanifold

We derive some important geometric identities for Lagrangian submanifolds immersed in a Kähler manifold and prove that there exists a canonical way to deform a Lagrangian submanifold by a parabolic flow through a family of Lagrangian submanifolds if the ambient space is a Ricci-flat Calabi-Yau manifold.

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Differential Geometry

A closed form for unitons

Unitons, i.e.\ harmonic spheres in a unitary group, correspond to \lq uniton bundles\rq, i.e.\ holomorphic bundles over the compactified tangent space to the complex line with certain triviality and other properties. In this paper, we use a monad representation similar to Donaldson's representation of instanton bundles to obtain a simple formula for the unitons. Using the monads, we show that real triviality for uniton bundles is automatic. We interpret the uniton number as the `length' of a jumping line of the bundle, and identify the uniton bundles which correspond to based maps into Grassmannians. We also show that energy- 3 unitons are 1 -unitons, and give some examples.

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Differential Geometry

A compact symmetric symplectic non-Kaehler manifold

In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler. Delzant's classification of 2n -dimensional symplectic manifolds with Hamiltonian action of n -dimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler. An example in this paper show that not all compact symplectic manifolds that admit Hamiltonian torus actions are Kaehler. Similar technique allows us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.

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Differential Geometry

A discrete version of the Darboux transform for isothermic surfaces

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all isothermic surfaces: they can be characterized by the fact that their parallel constant mean curvature surfaces are Christoffel and Darboux transforms at the same time. This characterization is used to define discrete nets of constant mean curvature. Basic properties of discrete nets of constant mean curvature are derived.

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Differential Geometry

A fake smooth CP^2 # RP^4

We show that the manifold *CP^2 # *RP^4, which is homotopy equivalent but not homeomorphic to CP^2 # RP^4, is in fact smoothable.

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Differential Geometry

A frame bundle generalization of multisymplectic field theories

This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generalized symplectic structure from the full frame bundle. The geometric structure of the vertically adapted frame bundle admits vector-valued field observables and produces vector-valued Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We show that the linear and affine multivelocity spaces and multiphase spaces for geometric field theories are associated to the vertically adapted frame bundle. In addition, the new geometry not only generalizes both the linear and the affine models of multisymplectic geometry but also resolves fundamental problems found in both multisymplectic models.

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Differential Geometry

A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications

Using the Bryant representation, we define a new flux on homology classes of CMC-1 surfaces in hyperbolic 3-space, satisfying a balancing formula which is useful to show nonexistencd of certain kinds of complete CMC-1 surfaces.

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Differential Geometry

A remark on Berezin's quantization and cut locus

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and two-point functions are defined when one variable does not belong to the cut locus of the other one.

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Differential Geometry

A remark on positively curved manifolds of dimensions 7 and 13

Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.

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