Multiple integrals in the calculus of variations and nonlinear elliptic systems

Weakly Differentiable Functions

Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ)

We study sequences $f_k:\Sigma_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq \Lambda$. Assume that $\Sigma_k$ converges to $\Sigma$ in moduli space, i.e. $\phi_k^\ast(\Sigma_k) \to \Sigma$ as complex structures for diffeomorphisms $\phi_k$. Then we construct a branched conformal immersion $f:\Sigma \to \R^n$ and M\"obius transformations $\sigma_k$, such that for a subsequence $\sigma_k \circ f_k \circ \phi_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely many points. For $\Lambda < 8\pi$ the map $f$ is unbranched. If the $\Sigma_k$ diverge in moduli space, then we show $\liminf_{k \to \infty} {\cal W}(f_k) \geq \min(8\pi,\omega^n_p)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2012/0020/0002/CAG-2012-0020-0002-a004.pdf

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Papers by: S.H. Abed, O. Amici, A. Asada, V.V. Astrhancev, S. Bacso, J.K. Beem, A. Bejancu, K. Belteky, I.M. Benn, L. Biesk, T.Q. Binh, N. Blazic, N. Bokan, K. Buchner, B. Casciaro, A.V. Chakmazjan, I. Comic, A. Crumeyrolle, I. Dimitric, C.T.J. Dodson, P. Dragila, S. Dragomir, J.J. Duistermaat, P.E. Ehrlich, J. Eichhorn, M. Falcitelli, M. Ferraris, S. Formella, M. Francaviglia, S.I. Goldberg, A. Gray, T. Hangan, M. Husty, G. Ivan, A. Jakubowicz, S. Janeczko, Z. Jankovic, J. Klein, F.C. Klepp, I. Kolar, Z. Kovacs, L. Kozma, B. Lukacs, V.S. Malakhovsky, K.B. Marathe, S. Markvorsen, I. Mihai, J. Mikesh, R. Miron, E. Molnar, A.M. Naveira, J. Nikic, J. Novotny, V. Oliker, T. Otsuki, A.M. Pastore, Z. Perjes, J.F. Pommaret, M. Puta, H. Reckziegel, A.H. Rocamora, O. Roschel, H. Sachs, A. Sebestyen, U. Simon, H. Singh, G.A.J. Sparling, P. Stavre, S. Steiner, R. Sulanke, J. Szenthe, J. Szilasi, L. Tamassy, E. Teufel, F. Tricerri, R.W. Tucker, C. Udriste, I. Vaisman, L. Vanhecke, L. Verstraelen, V.A. Vujicic, P.G. Walczak, H. Yasuda, A. Zajtz, V. Zoller

Topics in Differential Geometry

The Teichmuller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.

Asymptotics of the Teichm\"uller harmonic map flow

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.

/pdf/energy-quantization-for-a-singular-super-liouville-boundary-4x0b3frke4.pdf

Energy quantization for a singular super-Liouville boundary value problem

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension less than 6, partial regularity holds for stationary solutions with respect to general gravitino fields.

Partial Regularity for a Nonlinear Sigma Model with Gravitino in Higher Dimensions

Let $$(\phi _\alpha , \psi _\alpha )$$
 be a sequence of $$\alpha $$
 -Dirac-harmonic maps from a Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy. If the target N is a sphere $$S^{K-1}$$
 , we show that the energy identity and necklessness hold during the interior blow-up process as $$\alpha \searrow 1$$
 for such a sequence .

Energy identity and necklessness for $$\alpha $$ α -Dirac-harmonic maps into a sphere

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 $$\alpha $$
 -YMH fields as $$\alpha \rightarrow 1$$
 . For $$\alpha >1$$
 , some regularity results for $$\alpha $$
 -YMH field are shown. This is achieved by showing a regularity theorem for more general coupled systems, which extends the classical results of Ladyzhenskaya–Ural’ceva and Morrey.

Miaomiao Zhu

Papers

Asymptotics of the Teichm\"uller harmonic map flow

Energy quantization for a singular super-Liouville boundary value problem

Partial Regularity for a Nonlinear Sigma Model with Gravitino in Higher Dimensions

Energy identity and necklessness for $$\alpha $$ α -Dirac-harmonic maps into a sphere

The boundary value problem for Yang–Mills–Higgs fields