Conformal and semi-conformal biharmonic maps

We show that a conformal mapping between Riemannian manifolds of the same dimension n ≥ 3 is biharmonic if and only if the gradient of its dilation satisfies a certain second-order elliptic partial differential equation. On an Einstein manifold solutions can be generated from isoparametric functions. We characterise those semi-conformal submersions that are biharmonic in terms of their dilation and the fibre mean curvature vector field.      

Neck analysis for biharmonic maps

In this paper, we study the blow up of a sequence of (both extrinsic and intrinsic) biharmonic maps in dimension four with bounded energy and show that there is no neck in this process. Moreover, we apply the method to provide new proofs to the removable singularity theorem and energy identity theorem of biharmonic maps.     

Existence and nonexistence results for critical growth biharmonic elliptic equations

We prove existence of nontrivial solutions to semilinear fourth order problems at critical growth in some contractible domains which are perturbations of small capacity of domains having nontrivial topology. Compared with the second order case, some difficulties arise which are overcome by a decomposition method with respect to pairs of dual cones. In the case of Navier boundary conditions, further technical problems have to be solved by means of a careful application of concentration compactness lemmas. The required generalization of a Struwe type compactness lemma needs a somehow involved discussion of certain limit procedures. Also nonexistence results for positive solutions in the ball are obtained, extending a result of Pucci and Serrin on so-called critical dimensions to Navier boundary conditions. A Sobolev inequality with optimal constant and remainder term is proved, which is closely related to the critical dimension phenomenon. Here, this inequality serves as a tool in the proof of the existence results and in particular in the discussion of certain relevant energy levels.Received: 18 April 2002, Accepted: 3 September 2002, Published online: 17 December 2002Mathematics Subject Classification (2000): 35J65; 35J40, 58E05The first author was supported by MURST project "Metodi Variazionali ed Equazioni Differenziali non Lineari".     

On biharmonic maps and their generalizations

We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.Received: 6 February 2003, Accepted: 12 March 2003, Published online: 16 May 2003Mathematics Subject Classification (2000): 35J60, 35H20Pawel Strzelecki: Current address (till September 2003): Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany (email: strzelec@math.uni-bonn.de). The author is partially supported by KBN grant no. 2-PO3A-028-22;he gratefully acknowledgesthe hospitality of his colleagues from Bonn,and the generosity of Humboldt Foundation.     

The stress-energy tensor for biharmonic maps

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map. S. Montaldo was supported by PRIN-2005 (Italy): Riemannian Metrics and Differentiable Manifolds. C. Oniciuc was supported by a CNR-NATO (Italy) fellowship and by the Grant CEEX, ET, 5871/2006 (Romania).     

Partial regularity for biharmonic maps, revisited

Extending our previous results with Tristan Rivière for harmonic maps, we show how partial regularity for stationary biharmonic maps into arbitrary targets can be naturally obtained via gauge theory in any dimensions m ≥ 4.     

A monotonicity formula for stationary biharmonic maps

We give a rigorous proof of the monotonicity formula of S.-Y.A. Chang, L. Wang and P. Yang [3] for (extrinsically) stationary biharmonic maps of class W^2,2. This work was partially supported by SNF 200021-101930/1.     

On existence and nonexistence of some harmonic maps

LetS ^1.1 denote the submanifold of Lorentz spaceR ^2.1, which is composed of all pointsl withl ²=1 and letT ^1.1=R ^1.1/Z×Z. In this paper we study the existence of nontrivial harmonic maps fromT ^1.1 toS ^1.1 andH ², and construct a harmonic map for any homotopy class of maps fromT ² toS ^1.1.     

Large solutions for biharmonic maps in four dimensions

We seek critical points of the Hessian energy functional , where or Ω is the unit disk in and u : Ω → S ⁴. We show that has a critical point which is not homotopic to the constant map. Moreover, we prove that, for certain prescribed boundary data on ∂B, E _B achieves its infimum in at least two distinct homotopy classes of maps from B into S ⁴. The author was partially supported by SNF 200021-101930/1.     

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Boundary partial regularity for a class of biharmonic maps

We consider the Dirichlet problem for biharmonic maps u from a bounded, smooth domain ${\Omega\subset\mathbb R^n (n\ge 5)}$ to a compact, smooth Riemannian manifold ${N\subset{\mathbb {R}}^l}$ without boundary. For any smooth boundary data, we show that if u is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset ${\Sigma\subset\overline{\Omega}}$ , with ${H^{n-4}(\Sigma)=0}$ , such that ${\displaystyle u\in C^\infty(\overline\Omega\setminus\Sigma, N)}$ .     

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For and , we show that any minimizing biharmonic map from to S^k is smooth off a closed set whose Hausdorff dimension is at most n-5. When n = 5 and k = 4, for a parameter we introduce a -relaxed energy of the Hessian energy for maps in so that each minimizer of is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of for .Received: 5 April 2004, Accepted: 19 October 2004, Published online: 10 December 2004     

Energy identity of approximate biharmonic maps to Riemannian manifolds and its application

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac{4}{3}$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on ${\mathbb{R}}^4$. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps at time infinity.     

A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.