We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin--Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.
We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We...
We study quantitative stability for the fractional Yamabe functional introduced by Gonz\'alez and Qing. Its minimizers correspond to conformal metrics with constant fractional curvature. We prove that the deficit of the fractional Yamabe functional controls, with a suitable power, the distance to the set of minimizers....
We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved...
Shi-Bing Chen, M. Ghomi, Peng Wang· 3 citations· ⚡1
We study a two-parameter deformation of scalar curvature associated with the Gauduchon connections on compact Hermitian manifolds. A central identity expresses its dependence on the connection parameter through the squared norm of the trace-free part of the torsion of the Chern connection. We establish existence criter...
We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequal...
The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to E...
M. Bhakta, D. Ganguly, D. Karmakar· 0 citations
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