Broadly speaking my research focuses on partial differential equations of Elliptic and Parabolic type and non linear analysis on manifolds. To be more precise :
As a graduate student I started working on nonlinear elliptic partial differential equations and variational methods. Several problems arising in differential geometry, physics and other topics give rise to semi-linear variational elliptic equations. Particularly my aim is to study these equations on the hyperbolic space. It includes the study of existence, non existence and multiplicity of solutions of elliptic equations with critical Sobolev growth. I am also interested in the qualitative properties like symmetry, compactness, non degeneracy and stability of solutions for the non compact variational problems.
With time my interests have branched towards functional and geometric inequalities, spectral theory and heat kernel estimates. It includes the study of inequalities with sharp constants, optimal improvement of functional inequalities of Hardy and Poincare-type on Cartan-Hadamard manifolds.
Criticality theory for elliptic second order partial differential operator and Liouville Theorem for Schroedinger operator. Another aspect of my research is to study equivalence of heat kernels and Green functions on Riemannian manifolds for the second order operator.