On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

Uniqueness of Subelliptic Harmonic Maps

Let R^m be an open set, Nⁿ a Riemannian manifold, X a collection of vector fields on , and f a smooth map from into Nⁿ. We call f a subelliptic harmonic map if it is a critical point of the energy functional with respect to X. In this paper, we calculate the first and the second variations of the energy functional, and use them to prove the partial uniqueness of a subelliptic harmonic map under the condition that Nⁿ has the non-positive curvature. Then, we utilize the maximum principle for subelliptic PDEs to verify the global uniqueness of a subelliptic harmonic map under some other conditions.     

A Böchner technique for harmonic mappings from a 3-manifold to a surface

Let be a harmonic mapping from a Riemannian 3-manifold to a Riemannian 2-manifold. A smooth function on M is associated to , derived from the eigenvalues of the first fundamental form, the vanishing of which is equivalent to being a harmonic morphism. The Laplacian of this function is computed and a maximum principle applied to derive criteria when a harmonic map must be a harmonic morphism.     

A Weierstrass type representation for minimal surfaces in Sol

The normal Gauss map of a minimal surface in the model space of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

     

On the biharmonic and harmonic indices of the Hopf map

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map and modify it into a nonharmonic biharmonic map . We show to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

     

Nonlinear Markov Operators,Discrete Heat Flow,and Harmonic Maps Between Singular Spaces

We develop a theory of harmonic maps f:MN between singular spaces M and N. The target will be a complete metric space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov. The domain will be a measurable space (M,) with a given Markov kernel p(x,dy) on it. Given a measurable map f:MN, we define a new map Pf:MN in the following way: for each xM, the point Pf(x)N is the barycenter of the probability measure p(x,f ^–1(dy)) on N. The map f is called harmonic on DM if Pf=f on D. Our theory is a nonlinear generalization of the theory of Markov kernels and Markov chains on M. It allows to construct harmonic maps by an explicit nonlinear Markov chain algorithm (which under suitable conditions converges exponentially fast). Many smoothing and contraction properties of the linear Markov operator P _M,R carry over to the nonlinear Markov operator P=P _M,N . For instance, if the underlying Markov kernel has the strong Lipschitz Feller property then all harmonic maps will be Lipschitz continuous.     

Variationally harmonic maps with general boundary conditions: Boundary regularity

Let and be Riemannian manifolds, compact without boundary. We develop a definition of a variationally harmonic map with respect to a general boundary condition of the kind u(x)∊Γ(x) for a.e. , where are given submanifolds depending smoothly on x. The given definition of variationally harmonic maps is slightly more restrictive, but also more natural than the usual definition of stationary harmonic maps. After deducing an energy monotonicity formula, it is possible to derive a regularity theory for variationally harmonic maps with general boundary data. The results include full boundary regularity in the Dirichlet boundary case Γ(x) = {g(x)} for if does not carry a nonconstant harmonic 2-sphere.     

Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds

A ?ojasiewicz-type estimate is a powerful tool in studying the rigidity properties of the harmonic map heat flow. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flow from \(\mathbb{S}^{2}\) to \(\mathbb{S}^{2}\) (J. Differential Geom. 45 (1997), 593–610). In this note, using an inequality due to Sobolev, we will derive the same estimate for maps from \(\mathbb{S}^{2}\) to a compact Kähler manifold N with nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flow from \(\mathbb{S}^{2}\) to N, which generalizes Topping’s result.     

Sequences of harmonic maps in the 3‐sphere

We define two transforms of non‐conformal harmonic maps from a surface into the 3‐sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3‐sphere, H‐surfaces in Euclidean 3‐space and almost complex surfaces in the nearly Kähler manifold . As a consequence we can construct sequences of H‐surfaces and almost complex surfaces.     

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.     

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.     

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

We exhibit a stable finite time blowup regime for the 1‐corotational energy critical harmonic heat flow from ?² into a smooth compact revolution surface of ?³ that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy‐critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1‐equivariant energy critical wave map and Schrödinger map with ??² target by Merle, Raphaël, and Rodnianski. © 2012 Wiley Periodicals, Inc.     

On the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2-sphere

Let and be the unit disk and the unit sphere, and let be a radially symmetric harmonic map heat flow, whose singularities coincide with downward energy jumps. Then its finite time singularities are simple in the sense that precisely one harmonic sphere separates at a time.     

Subelliptic harmonic maps from Carnot groups

For subelliptic harmonic maps from a Carnot group into a Riemannian manifold without boundary, we prove that they are smooth near any -regular point (see Definition 1.3) for sufficiently small . As a consequence, any stationary subelliptic harmonic map is smooth away from a closed set with zero H^Q-2 measure. This extends the regularity theory for harmonic maps (cf. [SU], [Hf], [El], [Bf]) to this subelliptic setting.Received: 24 April 2002, Accepted: 30 September 2002, Published online: 17 December 2002Mathematics Subject Classification (2000): 35B65, 58J42     

Construction of Markov processes and associated multiplicative functionals from given harmonic measures

Summary LetE be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures onE that behave like harmonic measures associated with all relatively compact open sets inE (i.e. that satisfy a certain consistency condition), one can construct a Markov process onE and a multiplicative functional with values in [0, ) such that the hitting distributions of the process inflated by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the spaceE equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).     

Biharmonic map heat flow into manifolds of nonpositive curvature

Let M ^m and N be two compact Riemannian manifolds without boundary. We consider the L ² gradient flow for the energy . If and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 58E20, 58J35     

Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric

Given an orientable hypersurface M of a Lie group with a bi-invariant metric we consider the map N : M ⁿ that translates the normal vector field of M to the identity, which is a natural extension of the usual Gauss map of hypersurfaces in Euclidean spaces; it is proved that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map. One may then conclude that M has constant mean curvature (cmc) if and only if N is harmonic; some other aplications to cmc hypersurfaces of are also obtained.     

Harmonic maps with finite total energy

We will give a criteria for a nonnegative subharmonic function with finite energy on a complete manifold to be bounded. Using this we will prove that if on a complete noncompact Riemannian manifold , every harmonic function with finite energy is bounded, then every harmonic map with finite total energy from into a Cartan-Hadamard manifold must also have bounded image. No assumption on the curvature of is required. As a consequence, we will generalize some of the uniqueness results on homotopic harmonic maps by Schoen and Yau.

     

Uniformly bounded maximal -disks, Bers space and harmonic maps

We characterize Bers space by means of maximal -disks. As an application we show that the Hopf differential of a quasiregular harmonic map with respect to strongly negatively curved metric belongs to Bers space. Also we give further sufficient or necessary conditions for a holomorphic function to belong to Bers space.