Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

Dirac-harmonic maps couple a second order harmonic map type system with a first nonlinear Dirac equation. We consider approximate Dirac-harmonic maps \(\{(\phi _n,\psi _n)\}\), that is, maps that satisfy the Dirac-harmonic system up to controlled error terms. We show that such approximate Dirac-harmonic maps defined on a Riemann surface, that is, in dimension 2, continue to satisfy the basic properties of blow-up analysis like the energy identity and the no neck property. The assumptions are such that they hold for solutions of the heat flow of Dirac-harmonic maps. That flow turns the harmonic map type system into a parabolic system, but simply keeps the Dirac equation as a nonlinear first order constraint along the flow. As a corollary of the main result of this paper, when such a flow blows up at infinite time at interior points, we obtain an energy identity and the no neck property.     

On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d ≥ 3. It is shown that, generically, singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved. © 2017 Wiley Periodicals, Inc.     

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.     

Selfsimilar expanders of the harmonic map flow

We study the existence, uniqueness, and stability of self-similar expanders of the harmonic map heat flow in equivariant settings. We show that there exist selfsimilar solutions to any admissible initial data and that their uniqueness and stability properties are essentially determined by the energy-minimising properties of the so-called equator maps.     

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36]. © 2021 Wiley Periodicals LLC.     

Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L²-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.     

Concentrated boundary data and axially symmetric harmonic maps

We examine the existence problem for harmonic maps between the three-dimensional ball and the two-sphere. We utilize results on the classification of harmonic maps into hemispheres and a result on the regularity of the weak limit of energy minimizers over the class of axially symmetric maps to establish the existence of asmooth harmonic extension for boundary data suitably “concentrated” away from the axis of symmetry. In addition, we establish convergence results for the harmonic map heat flow problem for suitably “concentrated” axially symmetric initial and boundary data.     

Geometric evolution equations in critical dimensions

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance.     

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     

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.     

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.     

Partial regularity for weak heat flows into spheres

In this paper we show that a weak heat flow of harmonic maps from a compact Riemannian manifold (possibly with boundary) into a sphere, satisfying the monotonicity inequality and the energy inequality, is regular off a closed set of m-dimensional Hausdorff measure zero.     

Equivariant harmonic maps into the sphere via isoparametric maps

By using concrete isoparametric maps we obtain some new equivariant harmonic maps between spheres and solve equivariant boundary value problems for harmonic maps from unit open ballB ^m+1 intoS ⁿ. Research partially supported by NNSFC, SFECC and ICTP.     

On regularizing-rate estimates for solutions to the heat flow with initial value problem

For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.     

New constructions of twistor lifts for harmonic maps

We show that given a harmonic map φ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a J ₂-holomorphic twistor lift of φ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.     

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     

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.     

{\mathcal{F}} -stability of self-similar solutions to harmonic map heat flow

Inspired by the work of Colding and Minicozzi II: ”Generic mean curvature flow I: generic singularities”, we explore the notion of generic singularities for the harmonic map heat flow. We introduce ${\mathcal{F}}$ -functional and entropy for maps from Euclidean spaces. The critical points of the ${\mathcal{F}}$ -functional are exactly the weakly self-similar solutions to the harmonic map heat flow. We define the notion of ${\mathcal{F}}$ -stability for weakly self-similar solutions. The ${\mathcal{F}}$ -stability can be characterized by the semi-positive definiteness of the Jacobi operator acting on a subspace of variation fields.     

Note on equivariant harmonic maps in complex projective spaces

We give a simple criterion for equivariant harmonic maps into complex projective spaces CP ⁿ . As an application of the criterion, we give examples of equivariant harmonic cylinders. We also give examples of non-equivariant harmonic cylinders as perturbations of equivariant harmonic cylinders.