We study the existence of harmonic maps and Dirac-harmonic from degenerating surfaces to non-positive curved manifold via scheme Sacks Uhlenbeck. By choosing a suitable sequence $\alpha$-(Dirac-)harmonic closed hyperbolic surface, we get convergence cleaner energy identity under uniformly bounded assumption. In this identity, there is no loss near punctures. As an application, obtain result abo...

In this paper, we present an efficient and robust technique for surface remeshing based on harmonic maps. We show how to ensure a one-to-one mapping for the discrete harmonic map and introduce a cubic representation of the geometry based on curved PN triangles. Topological and geometrical limitations of harmonic maps are also put to the fore and discussed. We show that, with the proposed approa...

For the Schrödinger flow from R2 × R+ to the 2-sphere S2, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blowup time (if any), and that they blow up if and only if the length scale of the neare...

1. The Main Steps of this Work 2. The Analysis of Ravel's Bolero 2.1. Music "Objects" 2.1.1. Rhythmic Objects 2.1.2. Melodic Objects 2.1.3. Harmonic Objects 2.1.4. Resolution Objects 2.2. Meta-Structures 2.3. The Music Objects' Map 2.4. Growing Rules 2.4.1. Dynamics and Duplication Rules 2.4.2. Harmonic Rules 2.4.3. Timbral Textures 3. The Formal Model of Ravel's Bolero 3.1. Working Hypotheses ...

The purpose of this note is to state without proof certain. results on polynomial approximation to harmonic functions whose proofs will appear elsewhere later. In the plane of the complex variable z = x + iy, a (real) function u(z) harmonic in the interior of the Jordan curve C and continuous in the corresponding closed region C can be represented in C by a uniformly convergent series of harmon...

We develop an efficient numerical method for analyzing second harmonic generation (SHG) in two-dimensional photonic crystals composed of nonlinear circular cylinders embedded in a linear background medium. Instead of solving the governing inhomogeneous Helmholtz equation for the second harmonic wave in the entire structure directly, we define and solve a locally generated second harmonic field ...

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold N which has an equator (example: the sphere). In dimension 3, this article presents a necessary and sufficient condition on N for the existence of a smooth self-similar blow up profile. More generally, we study the relation between the minimizing properties of the equator map for the Dirichlet ...

The use of harmonic maps has been spectacularly successful in proving rigidity (and superrigidity) results for non-positively curved Riemannian manifolds. This is witnessed for example by results of Sui [31], Sampson [26], Corlette [6], Gromov and Schoen [18], Jost and Yau [22], and Mok, Sui and Yeung [23]. All of which are based on the pioneering existence theorem of Eells and Sampson [13] and...

We prove the regularity of weak 1/2−harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2−harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.

We give a new proof of the existence ([HM], [Ren]) of a Jenkins-Strebel differential Φ on a Riemann surface R with prescribed heights of cylinders by considering the harmonic map from R to the leaf space of the vertical foliation of Φ, thought of as a Riemannian graph. The novelty of the argument is that it is essentially Riemannian as well as elementary; moreover, the harmonic maps existence t...