arXiv research

Sacks-Uhlenbeck's result on metric spaces expanded.

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

In this paper, we study an $α$-flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the $α$-flows is a weak solution to the harmonic map flow. By an application of the $α$-flow, we present a simple proof of an energy identity of a minimizing sequence in each homotopy class.

Stability of Morse index for Yang-Mills connections in 4D.

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries…

$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in …

The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

New method proves existence of constant mean curvature disks on convex surfaces.

We extend the well-known Sacks-Uhlenbeck energy gap result (1981) for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic m…

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…

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 th…

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 …

We show the existence of Yang--Mills--Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blow-up behavior of a sequence of Sacks-Uhlenbeck type $α$-YMH fiel…

Study of area minimizing surfaces in homotopy classes of maps.

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is appli…

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_ρ$ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation $ρ$ of $π_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that…