https://neckpinch.com/authors/robert-koirala/New arXiv papers by Robert Koirala, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2608.14478/arxiv:2608.14478Fri, 14 Aug 2026 00:00:00 GMTBennett Chow, Robert Koirala — We introduce and study a Fisher information metric g^F_τ associated to the conjugate heat kernel of a Ricci flow (Mⁿ,gₜ). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that g^F_τ is monotone in scale and satisfies g^F_τ≤ gₜ. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect gₜ−g^F_τ. This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities 0<g^F_τ<gₜ at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for φ-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to 0 to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's ε-regularity theorem.Ricci flowPerelman entropyHeat kernelStability & uniquenessEigenvalues & spectrum https://neckpinch.com/papers/2605.21193/arxiv:2605.21193Wed, 20 May 2026 00:00:00 GMTRobert Koirala — We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein–Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein–Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's L^p-Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's Hₙ-centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.Ricci flowPerelman entropyHeat kernelStability & uniquenessGeometric inequalities