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Jonas Hirsch

Publications and source records attributed to Jonas Hirsch.

At least 19 recordsLinked to original sources
arXiv2026

Stationary varifolds with singularities II

For every dimension $m\geq 3$, we show the existence of an open set $U\subset \mathbb R^{m+1}$, a smooth Riemannian metric $g$ on it, and an integral $m$-dimensional stationary varifold $V$ in $(U,g)$ with the following properties. $V$ has a flat tangent plane of multiplicity $2$ at an interior point $p$, it is a smooth immersed minimal surface in $U\setminus \{p\}$, and it has infinite topology in any neighborhood of $p$. The construction starts from an idea of three of the authors, who in \cite{DHS} tried to build a similar example with $C^{m-1, α}$ regularity for $g$. That previous attempt contained an important error, which has been overcome using a crucial suggestion of gpt-astra-6.

math.DG↗
arXiv2026

Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems

We prove the Lipschitz regularity of solutions for a wide class of generalized Bernoulli free boundary problems, in vectorial, higher order, and non-minimizing settings. Our method does not rely on notions of viscosity solutions or comparison methods, which allows us to reach the optimal regularity for Bernoulli-type problems associated to elliptic operators which do not satisfy any maximum principle, namely the elasticity, biharmonic and Stokes equation. With a similar method we obtain the optimal Lipschitz regularity for stationary, non-minimizing solutions of the standard Bernoulli problem in two dimensions.

math.AP↗
arXiv2026

Stationary and stable varifolds with singularities

We construct minimal $m$-dimensional immersions in $\R^{m+1}$, equipped with a $C^{1, α}$ metric, $α\in [0,1)$, with a sequence of \emph{catenoidal necks} or \emph{floating disks} converging to an isolated, multiplicity $2$, singular flat point.

math.DG↗
arXiv2026

Full flexibility of the Monge-Ampère system in codimension $d_*-d+1$

We prove that $\mathcal{C}^{1,α}$ solutions to the Monge-Ampère system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every Hölder exponent $α<1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k = 2d_*$ and based on ideas from [Källen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension $d=2$ and codimension $k=2$. The same proof scheme further yields local full flexibility of isometric immersions of $d$-dimensional Riemannian metrics into Euclidean space of dimension $d_* + 1$, generalizing the result in [Lewicka 2025] proved for $d=k=2$. By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension $(d+1)d_*-d+1$.

math.AP↗
arXiv2025

A note on the a.e. second-order differentiability of rank-one convex functions

In the Euclidean setting, the well-known Alexandrov theorem states that convex functions are twice differentiable almost everywhere. In this note, we extend this theorem to rank-one convex functions. Our approach is novel in that it draws more from viscosity techniques developed in the context of fully nonlinear elliptic equations. As a byproduct, the original Alexandrov theorem can essentially be reduced to the a.e. differentiability of one-dimensional monotone functions, as presented in the appendix.

math.AP↗
arXiv2025

Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem

We consider area minimizing $m$-dimensional currents $\mathrm{mod}(p)$ in complete $C^2$ Riemannian manifolds $Σ$ of dimension $m+1$. For odd moduli we prove that, away from a closed rectifiable set of codimension $2$, the current in question is, locally, the union of finitely many smooth minimal hypersurfaces coming together at a common $C^{1,α}$ boundary of dimension $m-1$, and the result is optimal. For even $p$ such structure holds in a neighborhood of any point where at least one tangent cone has $(m-1)$-dimensional spine. These structural results are indeed the byproduct of a theorem that proves (for any modulus) uniqueness and decay towards such tangent cones. The underlying strategy of the proof is inspired by the techniques developed by Leon Simon in "Cylindrical tangent cones and the singular set of minimal submanifolds" (J. Diff. Geom. 1993) in a class of multiplicity one stationary varifolds. The major difficulty in our setting is produced by the fact that the cones and surfaces under investigation have arbitrary multiplicities ranging from $1$ to $\lfloor \frac{p}{2}\rfloor$.

math.AP↗
arXiv2025

Nonunique tangent maps at isolated singularities of minimizing $p$-harmonic maps

The analysis of ``tangent maps'' at singular points of energy minimizing maps plays an important role in our understanding of the fine structure of the singular set. This note presents the first example of a minimizing (not just stationary) $p$-harmonic map with nonunique tangent maps at an isolated singularity. We construct a $n$-dimensional manifold $N$ such that for every admissible tuple $p< m\le n+2$, there exists a map from $B_1^m$ into $N$ that minimizes the $p$-energy, has an isolated singularity at the origin and admits a continuum of distinct tangent maps. The construction builds upon and extends B.~ White's example for $p=2$ in the stationary case.

math.AP↗
arXiv2025

Isometric Immersions and Weak Solutions to the Darboux Equation

We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in C^{1,θ}$ with $θ>1/2$, and show that the classical correspondence between solutions of the Darboux equation and isometric immersions remains valid in this regime. The key ingredient is an extension of the classical flatness criterion to Hölder continuous metrics, achieved via an analysis of a weak notion of Gaussian curvature.

math.AP↗
arXiv2025

A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent

Given any short immersion from an $n$-dimensional bounded and simply connected domain into $\mathbb{R}^{n+1}$ and any Hölder exponent $α<(1+n^2-n)^{-1}$, we construct a $C^{1, α}$ isometric immersion arbitrarily close in the $C^0$ topology. This extends the classical Nash--Kuiper theorem and shows the flexibility of $C^{1, α}$ isometric immersions beyond Borisov's exponent. In particular, for $n=2$, the regularity threshold aligns with the Onsager exponent $1/3$ for the incompressible Euler equations. Our proof relies on three novelties that allow for the cancellation of leading-order error terms in the convex integration scheme: a new corrugation ansatz, an integration by parts procedure, and an adapted algebraic decomposition of these errors.

math.AP↗
arXiv2024

Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

We study complete minimal surfaces in $\mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy $\mathcal{W}: =\frac{1}{4} \int|\vec H|^2$. In codimension one, we prove that the $\mathcal{W}$-Morse index for any inverted minimal sphere or real projective plane with $m$ such ends is exactly $m-3=\frac{\mathcal{W}}{4π}-3$. We also consider several geometric properties -- for example, the property that all $m$ asymptotic planes meet at a single point -- of these minimal surfaces and explore their relation to the $\mathcal{W}$-Morse index of their inverted surfaces.

math.DG↗
arXiv2024

Interior regularity for two-dimensional stationary $Q$-valued maps

We prove that $2$-dimensional $Q$-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions.

math.AP↗
arXiv2024

Index estimates for sequences of harmonic maps

In this paper we study upper and lower bounds of the index and the nullity for sequences of harmonic maps with uniformly bounded Dirichlet energy from a two-dimensional Riemann surface into a compact target manifold. The main difficulty stems from the fact that in the limit the sequence can develop finitely many bubbles. We obtain the index bounds by studying the limiting behavior of sequences of eigenfunctions of the linearized operator and the key novelty of the present paper is that we diagonalize the index form of the Dirichlet energy with respect to a bilinear form which varies with the sequence of harmonic maps and which helps us to show the convergence of the sequence of eigenfunctions on the weak limit, the bubbles and the intermediate neck regions. Finally, we sketch how to modify our arguments in order to also cover the more general case of sequences of critical points of two-dimensional conformally invariant variational problems.

math.DG↗
arXiv2023

$C^{1,\frac{1}{3}-}$ very weak solutions to the two dimensional Monge-Ampére equation

For any $θ<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Ampère equation with regularity $C^{1,θ}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.

math.AP↗
arXiv2023

Excess decay for minimizing hypercurrents mod $2Q$

We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $Σ$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt} (T)\setminus \mathrm{spt}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $Σ$. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a $C^{1,α}$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.

math.AP↗
arXiv2023

On the Lawson-Osserman conjecture

We prove that if $u : B_1 \subset \mathbb{R}^2 \rightarrow \mathbb{R}^n$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case.

math.AP↗
arXiv2023

Higher integrability for measures satisfying a PDE constraint

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} μ= σ, \] where $μ\in \mathcal{M}(Ω;V)$ and $σ\in \mathcal{M}(Ω;W)$ are vector measures and the polar $\frac{\mathrm{d} μ}{\mathrm{d} |μ|}$ is uniformly close to a convex cone of $V$ intersecting the wave cone of $\mathcal{A}$ only at the origin. More precisely, we prove local compensated compactness estimates of the form \[ \|μ\|_{\mathrm{L}^p(Ω')} \lesssim |μ|(Ω) + |σ|(Ω), \qquad Ω' \Subset Ω. \] Here, the exponent $p$ belongs to the (optimal) range $1 \leq p < d/(d-k)$, $d$ is the dimension of $Ω$, and $k$ is the order of $\mathcal{A}$. We also obtain the limiting case $p = d/(d-k)$ for canceling constant-rank operators. We consider applications to compensated compactness and {applications to the theory of} functions of bounded variation and bounded deformation.

math.AP↗