arXiv subjects

Robert Koirala

Publications and source records attributed to Robert Koirala.

11 recordsLinked to original sources

arXiv2026

Volume Growth under Positive Intermediate Curvature and a Ricci Lower Bound

Let $(M^n,g)$ be a complete Riemannian manifold with $\Ric\ge-kg$ and uniformly positive $m$-intermediate curvature in the sense of Brendle--Hirsch--Johne. We prove that the Fisher eigenvalue $λ_{n-m+1}$ is small, after heat averaging, below the curvature scale $k^{-1}$. Consequently, balls have polynomial volume growth of order $R^{m-1}$ for $R\le k^{-1/2}$. At larger scales we obtain the corresponding estimate with an exponential factor $\exp(C\sqrt k R)$, and an example shows that this factor is necessary.

math.DG↗

arXiv2026

The Fisher Metric of the Ricci Flow Heat Kernel

We introduce and study a Fisher information metric $g^F_τ$ associated to the conjugate heat kernel of a Ricci flow $(M^n,g_t)$. 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 $g^F_τ\le g_t$. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect $g_t-g^F_τ.$ This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities $0<g^F_τ<g_t$ 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 $\varepsilon$-regularity theorem.

math.DG↗

arXiv2026

Unbounded Topology of Nodal Sets of Harmonic Functions

For every integer $n\ge 3$, every $1\le \ell\le n-2$, and every sufficiently large integer $m$, we construct harmonic functions $u_{m,\ell}$ on the unit ball $B_1(0)\subset\mathbb{R}^n$ such that the frequency is bounded independently of $m$, every point of the nodal set $\{u_{m,\ell}=0\}\cap B_{1/2}(0)$ is regular, but the Betti numbers satisfy \begin{align*} b_\ell\bigl(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\bigr)\ge 2m. \end{align*} Thus bounded frequency, even together with regularity of the nodal set, does not imply a uniform topological bound. In particular, these examples give counterexamples to the claimed global Betti-number bound of Lin and Liu.

math.AP↗

arXiv2026

Sharp Gaussian Isoperimetry along a Ricci Flow

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 $H_n$-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.

math.DG↗

arXiv2026

Nested nodal loops of biharmonic functions

Given any $n\in\mathbb{N}$, we construct a real-valued biharmonic polynomial on $\mathbb{R}^2$ whose zero set contains a nest of $n$ smooth, disjoint topological loops, meaning that the $k$-th loop lies inside the domain bounded by the $(k+1)$-st loop for $k=1,\ldots,n-1$. The case $n=2$, i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.

math.AP↗

arXiv2026

Nested nodal loops for sums of Laplace eigenfunctions

We study nested loops in zero sets of sums of Laplace eigenfunctions on closed surfaces. In the real-analytic category, answering a question of Logunov, we prove a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. We show that this analyticity hypothesis is sharp: on a smooth sphere, a linear combination of eigenfunctions with eigenvalues $0$ and $2$ can have infinitely many rooted double nests. We also answer a question of Logunov and Nadirashvili by constructing a planar biharmonic function whose nodal set contains a double nest, and we prove a quantitative bound for entire biharmonic functions of polynomial growth. The biharmonic construction gives a nodal-set manifestation of the failure of the Boggio--Hadamard conjecture from the 1900s.

math.AP↗

arXiv2026

Equidistribution of Eigenfunctions of Quantum Cat Maps

We prove that the short-period eigenfunctions of quantum cat maps constructed by Kim and the author equidistribute on $\mathbb{T}^2$ in the sense of semiclassical measures. We also show that their logarithmically large $\ell^\infty$-norm is asymptotically concentrated on a bounded number of coordinates. Thus, for this explicit family, strong coordinate localization coexists with semiclassical equidistribution. These results confirm the behavior suggested by earlier numerical evidence of Kim and the author, and contrast with the scarring phenomena for short-period eigenfunctions observed by Faure, Nonnenmacher, and De Bièvre.

math.AP↗

arXiv2026

On a Problem of M. Kac on Laplace Distributions

We give counterexamples to a problem of M. Kac in the Scottish Book, which asks whether a certain nonlinear operation on two characteristic functions characterizes Laplace distributions, in analogy with the Cramér--Lévy theorem for Gaussian distributions. We then give an affirmative answer to a refined version of the problem. Finally, we develop a general framework for such characterization problems, construct generalized counterexamples, and pose some open questions.

math.PR↗

arXiv2025

Structure Theory of Parabolic Nodal and Singular Sets

We establish new estimates for the size and structure of the nodal set $\{u=0\}$ and the singular set $\{u=|\nabla u|=0\}$ of solutions $u$ to parabolic inequalities with parabolic Lipschitz coefficients. In particular, we show that almost all of the nodal and singular sets are covered by regular parabolic Lipschitz graphs with estimates, and that both sets satisfy parabolic Minkoswki estimates depending only on a doubling quantity at a point. Many of our results are new even for the heat equation on $\mathbb{R}^{n}\times \mathbb{R}$.

math.AP↗

arXiv2024

Bounds on Eigenfunctions of Quantum Cat Maps

We study $\ell^\infty$ norms of $\ell^2$-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Bièvre), we show that there exists a sequence of eigenfunctions $u$ with $\|u\|_{\infty}\gtrsim (\log N)^{-1/2}$. For general eigenfunctions we show the upper bound $\|u\|_\infty\lesssim (\log N)^{-1/2}$. Here the semiclassical parameter is $h=(2πN)^{-1}$. Our upper bound is analogous to the one proved by Bérard for compact Riemannian manifolds without conjugate points.

math.SP↗

arXiv2021

Notes on regularity of Anosov splitting

In these expository notes, we give a proof of regularity of Anosov splitting for Anosov diffeomorphisms in a torus. We also generalize the idea to higher dimensions and to Anosov flows.

math.DS↗