Seiki Nishikawa

Harmonic Maps in Complex Finsler Geometry

Given a smooth map from a compact Riemann surface to a complex manifold M, equipped with a strongly pseudoconvex Finsler metric F, we define a natural notion of the \(\overline \partial\)-energy of the map. A harmonic map is then defined to be a critical point of the \(\overline \partial\)-energy functional. Under the condition that F, is weakly Kahler, we obtain the second variation formula of the functional, and prove that any \(\overline \partial\)-energy minimizing harmonic map from a Riemann sphere to a weakly Kahler Finsler manifold M, of positive curvature is either holomorphic or antiholomorphic. Significance of complex Finsler metrics in the study of holomorphic vector bundles, in particular its relation with the Hartshorne conjecture in complex algebraic geometry, is represented. A brief overview of complex Finsler geometry is also provided.

Lectures on geometric variational problems

An Introduction to Geometric Variational Problems).- I The General Setting.- 1. A General Framework.- 2. A Rough Classification of Geometric Variational Problems.- 3. Different Points of View on Geometric Variational Problems and Their Uses.- II A Review of Geometric Variational Problems.- 1. The Eigenvalue Problem for the Laplacian on Functions.- 2. Harmonic Forms.- 3. Length and Energy of Curves.- 4. The Energy of Maps.- 5. Minimal Submanifolds.- 6. Yang-Mills Fields.- 7. The Total Scalar Curvature Functional.- 8. Some Other Non-Local Functionals of Riemannian Metrics.- III Symmetry Considerations, Topological Constraints, and Interactions with Physics.- 1. Symmetry Considerations.- 2. Topological Constraints.- 3. Interactions with Physics.- Geometry of Gauge Fields.- 1 Donaldson Invariant of 4 Manifolds.- 2 Basic Properties of the Moduli Space of ASD Connections.- 3 Casson Invariant and Gauge Theory.- 4 Floer Homology.- 5 Donaldson Invariant as Topological Field Theory.- 6 Gauge Theory on 4 Manifolds with Product End.- 7 Equivariant Floer Theory, Higher Boundary and Degeneration at Infinity.- Theorems on the Regularity and Singularity of Minimal Surfaces and Harmonic Maps.- LECTURE 1 Basic Definitions, and the ?-Regularity and Compactness Theorems.- 1 Basic Definitions and the ?-Regularity and Compactness Theorems.- 2 The ?-Regularity and Compactness Theorems.- LECTURE 2 Tangent Maps and Affine Approximation of Subsets of Rn.- 1 Tangent Maps.- 2 Properties of Homogeneous Degree Zero Minimizers.- 3 Approximation of Subsets of Rn by Affine Subspaces.- 4 Further Properties of sing u.- 5 Homogeneous Degree Zero ? with dim S(?) = n - 3.- LECTURE 3 Asymptotics on Approach to Singular Points.- 1 Significance of Unique Asymptotic Limits.- 2 Lojasiewicz Inequalities for the Energy.- 3 Proof of Theorem 1 of 1.- LECTURE 4 Recent Results on Rectifiability and Smoothness Properties of sing u.- 1 Statement of Main Theorems.- 2 Brief Discussion of Techniques.

Harmonic maps from Riemann surfaces into complex Finsler manifolds

Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the

Inverse spectral theory for Riemannian foliations and curvature theory

\bar{\partial}

Symmetry Considerations, Topological Constraints, and Interactions with Physics

-energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the

A Review of Geometric Variational Problems

\bar{\partial}

The General Setting

-energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kahler, we determine the second variation of the functional, and prove that any

CONFORMALLY FLAT HYPERSURFACES IN A CONFORMALLY FLAT RIEMANNIAN MANIFOLD

\bar{\partial}

The heat equation for Riemannian foliations

-energy minimizing harmonic map from the Riemann sphere to a weakly Kahler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.

On characteristic classes of riemannian, conformal and projective foliations

Inverse spectral theory addresses the question of which geometrical data of a Riemannian manifold (M, g) with some extra geometrical structure can be recovered or not recovered from the spectra of naturally associated differential operators.