The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds

For a sequence of approximate Dirac-harmonic maps from a closed spin Riemann surface into a stationary Lorentzian manifold with uniformly bounded energy, we study the blow-up analysis and show that the Lorentzian energy identity holds. Moreover, when the targets are static Lorentzian manifolds, we prove the positive energy identity and the no neck property.

     

Regularity theorems and energy identities for Dirac-harmonic maps

We study Dirac-harmonic maps from a Riemann surface to a sphere We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.The research of QC and JYL was partially supported by NSFC. QC was also partially supported by the FOK Yingtung Education Foundation.     

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps \((u_n,v_n)\) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form \(g_N -\beta dt^2\) for some Riemannian metric \(g_N\) and some positive function \(\beta \) on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.     

Dirac-harmonic maps from index theory

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.     

Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.     

Regularity for Dirac-harmonic map with Ricci type spinor potential

We study the regularity of Dirac-harmonic maps with a Ricci type spinor potential; as a byproduct, we also get a approximate compactness result for the related problems.     

Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

For a sequence of coupled fields $\{({?}_n,{\psi }_n)\}$ from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.     

Gauss maps of the Ricci-mean curvature flow

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms (Trans Am Math Soc 149: 569–573, 1970) proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang (Math Res Lett 10(2–3):287–299, 2003) extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.     

Energy Identities for Dirac-harmonic Maps

We prove that for a sequence of Dirac-harmonic maps from a compact Riemannian surface to a n dimensional compact Riemannian manifold N with uniformly bounded energy, the energy identities hold during the blow-up process.     

Dirac-harmonic maps

We introduce a functional that couples the nonlinear sigma model with a spinor field: In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem. The research of QC and JYL was partially supported by NSFC and Fok Yingtung Education Fundation. QC also thanks the Max Planck Institute for Mathematics in the Sciences for support and good working conditions during the preparation of this paper.     

Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps

Let be open and a smooth, compact Riemannian manifold without boundary. We consider the approximated harmonic map equation for maps , where . For , we prove H?lder continuity for weak solution s which satisfy a certain smallness condition. For , we derive an energy estimate which allows to prove partial regularity for stationary solutions of the heat flow for harmonic maps in dimension . Received: 7 May 2001; / in final form: 22 February 2002 Published online: 2 December 2002     

Nonlinear Dirac equations on Riemann surfaces

We have developed analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We have provided the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions.      

Regularity for weakly Dirac-harmonic maps to hypersurfaces

We prove that a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface is smooth. Supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.     

A structure theorem of Dirac-harmonic maps between spheres

For an arbitrary Dirac-harmonic map (φ,ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N. On the basis, we could clarify all of non-trivial Dirac-harmonic maps from S ² to S ².     

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

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 W ^1,2 × L ⁴ modulo bubbles compactness of a sequence of such maps. Supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.     

$\bar \partial $‐energy integral and uniqueness of harmonic maps

In this paper, we introduce a new energy functional, ‐energy functional, instead of the total energy functional to investigate the uniqueness of harmonic maps with respect to any given metric on the unit disk. Even in the setting that the Hopf differentials of harmonic maps are not integrable, certain uniqueness theorems of harmonic maps are obtained, which improve a result due to Markovi? and Mateljevi? in 1999. Moreover, a generalized energy‐minimizing property of harmonic maps is discussed. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ASYMPTOTIC BEHAVIOR OF HARMONIC MAPS FROM COMPLETE MANIFOLDS

51.IntroductionLetMbeanm-dimensionalcompletenoncompactRiemannianmanifold,pEM.P.LiandL.F.Turnintroducedin[8]avolumecomparisoncondition(VC)(seethedefinitionbelow).TheyobtainedsomeimportantanalyticpropertiesifMsatisfies(VC)andtheRiccicurvatureconditionRice(x)2--(m--1)K/(1 r(x))',wherer(x)isthedistancefromptox,K20isaconstant.ItisinterestingtoknowmoreabouttheanalysisonsuchamanifoldM.LetNbeacompleteRiemannianmanifoldwiththesectionalcurvatureKNboundedabovebysomeconstantK20.B.(T)denotestheg…     

The Gauss map of minimal surfaces in Berger spheres

It is proved that a pair of spinors satisfying a Dirac type equation represents surfaces immersed in Berger spheres with prescribed mean curvature. Using this, we prove that the Gauss map of a minimal surface immersed in a Berger sphere is harmonic. Conversely, we exhibit a representation of minimal surfaces in Berger spheres in terms of a given harmonic map. The examples we constructed appear in associated families.