Research

My current research lies in Riemannian geometry, with a focus on the classification of spaces admitting special curvature conditions. Broadly, I study the question: when does a smooth manifold admit a “best” metric? This problem, asks whether an arbitrary metric can be refined to one with richer symmetry or more rigid curvature. I approach it in the setting of Lie groups and homogeneous spaces, where geometric and algebraic structures interact transparently.

A central theme in my work is the study of m-quasi-Einstein metrics, which generalize Einstein and Ricci soliton metrics and appear naturally in the theory of conformal geometry, general relativity, and black hole physics. In my recent paper, I classified all nilpotent and unimodular solvable Lie groups that admit totally left-invariant quasi-Einstein metrics. The classification shows that the only nilpotent examples are Heisenberg groups, and in the solvable case I obtained an explicit structural description.

In my current project, I study conformally Einstein metrics on homogeneous spaces. Building on the work of Petersen and Wylie, I completed the classification of homogeneous conformally Einstein manifolds by proving that every simply connected, irreducible, non-locally conformally flat example is either solvable, compact, or a product of a compact Einstein space with Euclidean space. The argument uses tools from geometric invariant theory.

My collaborative work explores Ricci flow on homogeneous spaces, focusing on the long-time behavior of left-invariant metrics. We showed that every left-invariant metric on S^7 x S^7 evolves to one with positive Ricci curvature under the normalized Ricci flow, and we hope to extend this to products of other homogeneous spaces.