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Jesse Ratzkin

Publications and source records attributed to Jesse Ratzkin.

At least 19 recordsLinked to original sources
arXiv2026

Existence of Yamabe stability optimizers

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.

math.DG↗
arXiv2026

Quantitative stability for the Trudinger-Moser inequality

We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if either (i) the exponential rate of growth is sufficiently small or (ii) the domain is a round disk. The latter estimate remains valid even in the critical case. Both proofs rely on a new spectral gap that we prove, which may be of independent interest. Additionally we show that the same stability estimate holds in the nondegenerate case, and that this occurs generically.

math.AP↗
arXiv2026

Weighted Hardy-Sobolev type inequalities with boundary terms

In this paper we establish a new class of weighted Hardy-Sobolev type inequalities under mild monotonicity assumptions on the weight function. As a consequence, we derive the corresponding weighted Sobolev and trace-type inequalities. These results play an important role in the analysis of elliptic problems with Neumann or Robin boundary conditions in unbounded domains.

math.AP↗
arXiv2025

Classification of fractional, singular Yamabe metrics on a twice punctured sphere I

The Delaunay metrics form a family of conformally flat, constant fractional Q-curvature metrics on a twice-punctured sphere. They are all (after a M\"obius transformation) rotationally symmetric and periodic, and admit several elegant variational descriptions. We prove that, when s is close to but less than 1, any complete, conformally flat constant Q-curvature metric on a twice-punctured sphere is a Delaunay metric. Along the way, we prove a sharp a priori bound for the conformal factor of these metrics, which may be of independent interest.

math.DG↗
arXiv2025

An end to end gluing construction for metrics of constant Q-curvature

We produce many new complete, constant Q-curvature metrics on finitely punctured spheres by gluing together known examples. In our construction we truncate one end of each summand and glue the two summands together "end-to-end," where we've truncated them. We use this construction to show that the unmarked moduli space of solutions with a fixed number of punctures is topologically nontrivial provided the number of punctures is at least four.

math.DG↗
arXiv2024

Quantitative stability of the total $Q$-curvature near minimizing metrics

Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \leq k < \frac{n}{2}$. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the $Q$-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case $k=1$. Next we prove, for any integer $1 \leq k< \frac{n}{2}$, the existence of an $n$-dimensional Riemannian manifold such that the $k$-th order $Q$-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field.

math.AP↗
arXiv2024

Moduli space theory for complete, constant Q-curvature metrics on finitely punctured spheres

We study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov-Hausdorff topology, has the local structure of a real analytic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighborhood in the moduli space is actually a real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighborhood of the moduli space embeds as a Lagrangian submanifold in the parameter space.

math.DG↗
arXiv2023

Compactness of singular solutions to the sixth order GJMS equation

We study compactness properties of the set of conformally flat singular metrics with constant, positive sixth order Q-curvature on a finitely punctured sphere. Based on a recent classification of the local asymptotic behavior near isolated singularities, we introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov--Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research.

math.DG↗
arXiv2022

Constant Q-curvature metrics with Delaunay ends: the nondegenerate case

We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.

math.DG↗
arXiv2021

Compactness within the space of complete, constant Q-curvature metrics on the sphere with isolated singularities

In this paper we consider the moduli space of complete, conformally flat metrics on a sphere with k punctures having constant positive Q-curvature and positive scalar curvature. Previous work has shown that such metrics admit an asymptotic expansion near each puncture, allowing one to define an asymptotic necksize of each singular point. We prove that any set in the moduli space such that the distances between distinct punctures and the asymptotic necksizes all remain bounded away from zero is sequentially compact, mirroring a theorem of D. Pollack about singular Yamabe metrics. Along the way we define a radial Pohozaev invariant at each puncture and refine some a priori bounds of the conformal factor, which may be of independent interest.

math.DG↗
arXiv2020

Foliation of an asymptotically flat end by critical capacitors

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace-Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the non-local nature of our problem

math.AP↗
arXiv2020

On a fourth order conformal invariant

In this note we prove that a fourth order conformal invariant on the product of a circle with an (n-1)-dimensional sphere can be arbitrarily close to that of the n-dimensional sphere, generalizing a result of Schoen about the classical Yamabe invariant.

math.DG↗
arXiv2020

On constant Q-curvature metrics with isolated singularities

In this paper we derive a refined asymptotic expansion, near an isolated singularity, for conformally flat metrics with constant positive Q-curvature and positive scalar curvature. The condition that the metric has constant Q-curvature forces the conformal factor to satisfy a fourth order nonlinear partial differential equation with critical Sobolev growth, whose leading term is the bilaplacian. We model our results on a similar asymptotic expansion for conformally flat, constant scalar curvature metrics proven by Korevaar, Mazzeo, Pacard, and Schoen. Along the way we analyze the linearization of the Q-curvature equation about the Delaunay metrics recently discovered by Frank and König, which may be of independent interest.

math.DG↗
arXiv2014

Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature

Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $Ω\subset M$ be a suitable domain, and let $λ(Ω)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $Ω$. We prove several bounds for the rate of decrease of $λ(Ω)$ and $Ω$ increases, and a result comparing the rate of decrease of $λ$ before and after a conformal diffeomorphism. Along the way, we prove a reverse-Holder inequality for the first eigenfunction, which generalizes results of Chiti to the monifold setting and may be of independent interest

math.AP↗
arXiv2014

A reverse Holder inequality for extremal Sobolev functions

Let $n \geq 2$, let $Ω\subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is $$\mathcal{C}_p(Ω) = \inf \left \{ \frac{\int_Ω|\nabla v|^2}{\left ( \int_Ω|v|^p \right )^{2/p}} \right \} = \frac{\int_Ω|\nabla u|^2}{\left ( \int_Ω|u|^p \right )^{2/p}}.$$ Our inequality has the form $\| u \|_{L^p} \geq K \| u \|_{L^q}$ for any $q > p$, where $K$ depends only on $n$, $p$, $q$, and $\mathcal{C}_p(Ω)$. This result generalizes work of Chiti, regarding the first Dirichlet eigenfunction of the Laplacian, and of van den Berg, regarding the torsion function.

math.AP↗
arXiv2014

A numerical investigation of level sets of extremal Sobolev functions

In this paper we investigate the level sets of extremal Sobolev functions for subcritical exponents p. We conjecture that as p increases the corresponding extremal functions become more peaked, which we can measure by comparing their distribution functions. Then we provide compelling numerical evidence for our conjecture.

math.NA↗
arXiv2012

An isoperimetric inequality for extremal Sobolev functions

Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm in terms of their p-norm. This inequality is sharp, and it is an equality if and only if the domain is a round ball. Our result generalizes a theorem of Payne and Rayner and our proof relies on integral rearrangements and an analysis of the ODE corresponding to the radial case.

math.AP↗