We discuss two topics in this talk. First we study compact Ricci-flat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) ...

Given a compact smooth manifold Mn without boundary and n ≥ 3, the Lp-norm of the curvature tensor, defines a Riemannian functional on the space of Riemannian metrics with unit volume M1. Consider ...

The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved ...

In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform ...

The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. ...

The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions ...

This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative ...

Cauchy problems for elliptic partial differential equations arise in many important applications, such as, cardiography, nondestructive testing, heat transfer, sonic boom produced by a maneuvering aerofoil, etc. Elliptic ...