Searcharxiv⌕ Search

arXiv subjects

Jonah A. J. Duncan

Publications and source records attributed to Jonah A. J. Duncan.

11 recordsLinked to original sources
arXiv2026

Properties of hypersurface singular sets of solutions to the $σ_k$-Yamabe equation in the negative cone

We consider conformally flat Lipschitz viscosity solutions to the $σ_k$-Yamabe equation in the negative cone which admit smooth hypersurface singularities. Under natural regularity assumptions (that are satisfied by solutions to the $σ_k$-Loewner-Nirenberg problem on annuli, for example), we first prove that the trace and normal derivatives of such a solution along the hypersurface satisfy a certain PDE. For $k=2$, we also show that the hypersurface is minimal with respect to the Lipschitz solution and address some questions related to the formal expansion of the solution near the hypersurface.

math.AP↗
arXiv2025

The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates

In this paper we give a complete classification of positive viscosity solutions $w$ to conformally invariant equations of the form \begin{align}\label{ab}\tag{$*$} \begin{cases} f(λ(-A_w)) = \frac{1}{2}, \quad λ(-A_w)\inΓ& \text{in }\mathbb{R}_+^n \newline w = 0 & \text{on }\partial\mathbb{R}_+^n, \end{cases} \end{align} where $A_w$ is the Schouten tensor of the metric $g_w = w^{-2}|dx|^2$, $Γ\subset\mathbb{R}^n$ is a symmetric convex cone and $f$ is an associated defining function satisfying standard assumptions. Solutions to \eqref{ab} yield metrics $g_w$ of negative curvature-type which are locally complete near $\partial\mathbb{R}_+^n$. In particular, when $(f,Γ) = (σ_1,Γ_1^+)$, \eqref{ab} is the Loewner-Nirenberg problem in the upper half-space. More precisely, let $μ_Γ^+$ denote the unique constant satisfying $(-μ_Γ^+, 1,\dots,1)\in\partialΓ$. We show that when $μ_Γ^+ >1$ (e.g. when $Γ= Γ_k^+$ for $k<\frac{n}{2}$), the hyperbolic solution $w^{(0)}(x) := x_n$ is the unique solution to \eqref{ab}. More surprisingly, we show that when $μ_Γ^+ \leq 1$ (e.g. when $Γ= Γ_k^+$ for $k\geq \frac{n}{2}$), the solution set consists of a monotonically increasing one-parameter family $\{w^{(a)}(x_n)\}_{a\geq 0}$, of which the hyperbolic solution $w^{(0)}$ is the minimal solution. In either case, solutions of \eqref{ab} are functions of $x_n$. Our proof involves a novel application of the method of moving spheres for which we must establish new estimates and regularity near $\partial\mathbb{R}_+^n$, followed by a delicate ODE analysis. As an application, we give counterexamples to local boundary $C^0$ estimates on solutions to the fully nonlinear Loewner-Nirenberg problem when $μ_Γ^+ \leq 1$.

math.AP↗
arXiv2025

The $σ_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with smooth non-empty boundary $\partial M$. Let $Γ\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $Γ$ satisfying standard assumptions. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u = u^{-2}g_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem \begin{align*} \begin{cases} f(λ(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad λ(-g_u^{-1}A_{g_u})\inΓ& \text{on }M\backslash \partial M \newline u = 0 & \text{on }\partial M \end{cases} \end{align*} admits a solution if $μ_Γ^+ > 1-δ$, where $μ_Γ^+$ is defined by $(-μ_Γ^+,1,\dots,1)\in\partialΓ$ and $δ>0$ is a constant depending on certain geometric data. In particular, we solve the $σ_k$-Loewner-Nirenberg problem for all $k\leq \frac{n}{2}$, which extends recent work of the authors to include the important threshold case $k=\frac{n}{2}$. In the process, we establish that the fully nonlinear Loewner-Nirenberg problem and corresponding Dirichlet boundary value problem with positive boundary data admit solutions if there exists a conformal metric $g\in[g_0]$ such that $λ(-g^{-1}A_g)\inΓ$ on $M$; these latter results require no assumption on $μ_Γ^+$ and are new when $(1,0,\dots,0)\in\partialΓ$.

math.AP↗
arXiv2024

Fully nonlinear Yamabe-type problems on non-compact manifolds

We obtain existence results for a class of fully nonlinear Yamabe-type problems on non-compact manifolds, addressing both the so-called positive and negative cases. We also give explicit examples of manifolds with warped product ends and asymptotically flat ends satisfying the hypotheses of our theorems.

math.AP↗
arXiv2023

The $σ_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with non-empty boundary $\partial M$. Let $Γ\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $Γ$ satisfying standard assumptions. Under an algebraic condition on $Γ$, which is satisfied for example by the Gårding cones $Γ_k^+$ when $k<\frac{n}{2}$, we prove the existence of a Lipschitz viscosity solution $g_u = e^{2u}g_0$ to the fully nonlinear Loewner-Nirenberg problem associated to $(f,Γ)$, \begin{align*} \begin{cases} f(λ(-g_u^{-1}A_{g_u})) = 1, \quad λ(-g_u^{-1}A_{g_u}) \in Γ& \mathrm{on~}M\backslash\partial M \newline u(x)\rightarrow+\infty & \mathrm{as~}\operatorname{dist}_{g_0}(x,\partial M)\rightarrow 0, \end{cases} \end{align*} where $A_{g_u}$ is the Schouten tensor of $g_u$. Previous results on Euclidean domains show that, in general, $u$ is not differentiable. The solution $u$ is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner-Nirenberg problems on approximating cones containing $(1,0,\dots,0)$, for which we also have uniqueness. In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest. An important feature of our paper is that the existence of a conformal metric $g$ satisfying $λ(-g^{-1}A_g)\inΓ$ on $M$ is a consequence of our results, rather than an assumption.

math.AP↗
arXiv2023

An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

Suppose that $Σ^n\subset\mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $λ_1$ of the induced Laplace-Beltrami operator on $Σ$ satisfies $λ_1 \geq \frac{n}{2}+ a_n(Λ^6 + b_n)^{-1}$, where $a_n$ and $b_n$ are explicit dimensional constants and $Λ$ is an upper bound for the length of the second fundamental form of $Σ$. This provides the first explicitly computable improvement on Choi & Wang's lower bound $λ_1 \geq \frac{n}{2}$ without any further assumptions on $Σ$.

math.DG↗
arXiv2023

Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates

We obtain a priori local pointwise second derivative estimates for solutions $u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any $W^{2,p}$ norm can be dropped. Our results are derived using integral estimates.

math.AP↗
arXiv2022

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let $g_0$ be a smooth Riemannian metric on a closed manifold $M^n$ of dimension $n\geq 3$. We study the existence of a smooth metric $g$ conformal to $g_0$ whose Schouten tensor $A_g$ satisfies the differential inclusion $λ(g^{-1}A_g)\inΓ$ on $M^n$, where $Γ\subset\mathbb{R}^n$ is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric $g_1$ conformal to $g_0$ satisfying $λ(g_1^{-1}A_{g_1})\in\bar{Γ'}$ in the viscosity sense on $M^n$, together with a nondegenerate ellipticity condition, where $Γ' = Γ$ or $Γ'$ is a cone slightly smaller than $Γ$. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the solvability of the $σ_2$-Yamabe problem is equivalent to positivity of a nonlinear eigenvalue for the $σ_2$-operator in three dimensions. We also give a generalisation of a theorem of Aubin and Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

math.AP↗