Research Projects

Functional and geometric inequalities such as the Sobolev, Poincaré, Logarithmic Sobolev, Hardy, and isoperimetric inequalities form the cornerstone of analysis and PDE theory. Their stability refers to quantitative estimates measuring how far a given function or set is from attaining equality in these inequalities. In this research project, we focus on the stability of a wide class of functional inequalities, including: 1. Poincar\'e–Sobolev inequalities. 2. Hardy-type inequalities. 3. Heisenberg uncertainty principle. 4. Gagliardo–Nirenberg–Sobolev interpolation inequalities. 5. Logarithmic Sobolev inequalities. We aim to investigate these inequalities in the setting of Riemannian manifolds with non-positive Ricci curvature, with particular emphasis on Cartan–Hadamard manifolds (complete, simply connected manifolds with non-positive sectional curvature). Understanding the quantitative stability of these inequalities is crucial for applications in: 1. The long-time behaviour and convergence rates of solutions to nonlinear PDEs. 2. The compactness and concentration phenomena in variational problems. 3. Geometric analysis and comparison geometry, where curvature conditions influence analytic properties. Furthermore, understanding stability has profound implications: it refines classical inequalities by identifying almost extremizers and allows control over perturbations, leading to sharp estimates crucial in analysing nonlinear PDEs and geometric variational problems. We are, in particular, interested in fast diffusion flows on the manifolds. We are specifically interested in the following questions: 1. We plan to derive sharp quantitative stability results for any dimension in the multi-bubble case. Given the counter-example constructed in our previous work [Bhakta-Ganguly-Karmakar-Mazumdar '2023] for N \geq 6, it is clear that the linear stability is not valid, although we can obtain some nonlinear stability in the spirit of [Deng-Sun-Wei, Duke Math Journal, '2024]. This is a very challenging problem, as we need to derive the optimal threshold for the exponent where the linear stability fails. Although we have derived all the necessary ingredients, in particular, the optimal interaction estimates in our previous works on the stability of Poincaré-Sobolev inequalities on the hyperbolic space, to complete the non-linear (sharp) stability estimates for the Poincaré-Sobolev inequalities. 2. In the same spirit, we plan to derive sharp quantitative stability results for the Heisenberg Uncertainty principle, Log-Sobolev inequality, Poincar\'e inequality with Gaussian measure in Cartan-Hadamard Manifolds. In this regard, in a joint work with [Do-Ganguly-Lam-Lu, '2024, we already obtained weighted stability for the Heisenberg Uncertainty principle and non-optimal Log-Sobolev inequalities on the hyperbolic space. 3. On the existence of positive radial solutions to the fast diffusion equation with critical Sobolev exponent on the hyperbolic space.