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Tobias Lamm

Publications and source records attributed to Tobias Lamm.

At least 19 recordsLinked to original sources
arXiv2025

Sacks-Uhlenbeck type regularity for subcritical generalized $p$-harmonic maps into Homogeneous targets

Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain dimension $n$, we show a uniform $C^{1,α}$-regularity result for a sequence of such maps in the limit $p \searrow n$, assuming a uniform $n$-energy bound on its elements. The method of the proof follows the exact same lines as in \cite{strz3} but we need to check uniformity of estimates not previously considered there.

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arXiv2025

Higher dimensional Sacks-Uhlenbeck-type functionals and applications

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new null-homotopic such maps. The proof follows a similar perturbative argument, which in high dimensions leads to a degenerate and double-phase-type Euler-Lagrange system, making the uniform regularity needed to formalize the bubbling harder to achieve. Then, we develop a refined neck-analysis leading to an energy identity along the approximation, assuming a suitable Struwe-type entropy bound along a sequence of critical points. Finally, we combine these results to solve quite general min-max problems for the $n$-energy modulo bubbling.

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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.

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arXiv2023

Ricci flow of $W^{2,2}$-metrics in four dimensions

In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values $g$ are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to $W^{2,2}$ and satisfy $ \frac{1}{a}h\leq g\leq a h$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution. Results for a related non-compact setting are also presented. Various $L^p$ estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature $\geq k$ for $W^{2,2}$ metrics $g$ on closed four manifolds which are bounded in the $L^{\infty}$ sense by $ \frac{1}{a}h\leq g\leq a h$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$.

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arXiv2023

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space $\mathbb{R}^d$ and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space $\dot{B}^{\frac{d}{p}}_{p,\infty}(\mathbb{R}^d)$ where $d<p<\infty$. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form $\|\nabla u(t) \|_{L^\infty(\mathbb{R}^d)}\leq Ct^{-\frac12}$. Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.

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arXiv2022

Rigidity of $\varepsilon$-harmonic maps of low degree

In 1981, Sacks and Uhlenbeck introduced their famous $α$-energy as a way to approximate the Dirichlet energy and produce harmonic maps from surfaces into Riemannian manifolds. However, the second and third authors together with Malchiodi ([11],[12]) showed that for maps between two-spheres this method does not capture every harmonic map. They established a gap theorem for $α$-harmonic maps of degree zero and also showed that below a certain energy bound $α$-harmonic maps of degree one are rotations. We establish similar results for $\varepsilon$-harmonic maps $u_\varepsilon \colon S^2\rightarrow S^2$, which are critical points of the $\varepsilon$-energy introduced by the second author in [9]. In particular, we similarly show that $\varepsilon$-harmonic maps of degree zero with energy below $8π$ are constant and that maps of degree $\pm 1$ with energy below $12π$ are of the form $Rx$ with $R\in O(3)$. Moreover, we construct non-trivial $\varepsilon$-harmonic maps of degree zero with energy $> 8π$.

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arXiv2020

A gap theorem for $α$-harmonic maps between two-spheres

In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $α$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $α$-harmonic maps of degree $-1, 0$ or $1$.

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arXiv2020

Existence of expanders of the harmonic map flow

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded as a hypersurface of some Euclidean space and the initial condition is a Lipschitz map that is homotopic to a constant.

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arXiv2020

Biharmonic wave maps: Local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.

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arXiv2019

Biharmonic wave maps into spheres

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.

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arXiv2019

Reflection of Willmore surfaces with free boundaries

We study immersed surfaces in $\mathbb{R}^3$ which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

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arXiv2019

Local foliation of manifolds by surfaces of Willmore type

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature.

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arXiv2015

Conformal Willmore Tori in $\mathbb{R}^4$

For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4πk$. Moreover, we show that the energy value $8π$ cannot be attained by such an immersion. Additionally, we characterize the branched double covers $T^2\to S^2 \times \{0\}$ as the only branched conformal immersions, up to Möbius transformations of $\mathbb{R}^4$, from a torus into $\mathbb{R}^4$ with at least one branch point and Willmore energy $8π$. Using a perturbation argument in order to regularize a branched double cover, we finally show that the infimum of the Willmore energy in every conformal class of tori is less than or equal to $8π$.

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arXiv2015

Global estimates and energy identities for elliptic systems with antisymmetric potentials

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular our results confirm a conjecture of Rivière in the two-dimensional setting.

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arXiv2015

Limits of $α$-harmonic maps

Critical points of approximations of the Dirichlet energy à la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of $S^2$ are the only critical points of $E_α$ for maps from $S^2$ to $S^2$ whose $α$-energy lies below some threshold. In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of $α$-harmonic maps.

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arXiv2015

Rigidity and non-rigidity results for conformal immersions

In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in $\mathbb{R}^n$ with $n\ge 4$. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a blow-up process. For conformal immersions of spheres with large enough Willmore energies, we construct explicit counterexamples to a quantitative rigidity result and this complements the recently obtained rigidity results in [LaNg13].

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