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

https://www.mis.mpg.de/preprints/2009/preprint2009_51.pdf

Some explicit constructions of Dirac-harmonic maps

/pdf/a-global-weak-solution-of-the-dirac-harmonic-map-flow-1dv6zq1olm.pdf

A global weak solution of the Dirac-harmonic map flow

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac equations with such boundary conditions. We also apply these estimates to a new nonlinear elliptic-parabolic problem, the Dirac-harmonic heat flow on Riemannian spin manifolds. This problem is motivated by the supersymmetric nonlinear σ-model and combines a harmonic heat flow type equation with a Dirac equation that depends nonlinearly on the flow.

/pdf/estimates-for-solutions-of-dirac-equations-and-an-3h0yyeo9s1.pdf

Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem

We study Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy and show the so-called generalized energy identity in the case that the domain converges to a spin surface with only Neveu–Schwarz type nodes. We find condition that is both necessary and sufficient for the W1,2 × L4 modulo bubbles compactness of a sequence of such maps.

/pdf/dirac-harmonic-maps-from-degenerating-spin-surfaces-i-the-23reso5hjr.pdf

Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler–Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Riviere’s regularity theory and Riesz potential theory.

/pdf/regularity-of-solutions-of-the-nonlinear-sigma-model-with-2zolnyvgyl.pdf

Miaomiao Zhu

Papers

Some explicit constructions of Dirac-harmonic maps

A global weak solution of the Dirac-harmonic map flow

Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem

Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case

Regularity of Solutions of the Nonlinear Sigma Model with Gravitino