Spectral Notions for Conformal Maps: a Survey (original) (raw)

Conformal Spectrum and Harmonic maps

Eprint Arxiv 1007 3104, 2010

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that the map associating a finite number of eigenvectors of the maximizing lambda_1\lambda_1lambda_1 into the sphere is harmonic, establishing a link between conformal spectrum and harmonic maps.

On absolutely conformal mappings

Publicationes Mathematicae Debrecen, 2010

Let Ω be a domain in R n. It is proved that, if u ∈ C 1 (Ω; R n) and there holds the formula ∇u(x) n = n n/2 | det ∇u(x)| in Ω, then for n ≥ 3 u is a restriction of a Möbius transformation, and for n = 2, u is an analytic function. This extends, partially, the well-known Liouville theorem ([6]), wich states that if u ∈ ACL n (Ω; R n), n ≥ 3, and the condition ∇u(x) n = n n/2 det ∇u(x) is satisfied a.e. in Ω, then u is a restriction of a Möbius transformation.

A note on convex conformal mappings

Proceedings of the American Mathematical Society, 2018

We establish a new characterization for a conformal mapping of the unit disk D to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of λ of the Poincaré metric λ|dw| of a convex domain are strictly convex.

On conformal, harmonic mappings and Dirichlet’s integral

2011

This paper has an expository character, however we present as well some new results and new proofs. We prove a complex version of Dirichlet's principle in the plane and give some applications of it as well as estimates of Dirichlet's integral from below. Some of the results in the plane are generalized to higher dimensions. Roughly speaking, under the appropriate conditions we estimate the n-Dirichlet integral of a mapping u defined on a domain Ω ⊂ R n , n ≥ 2, by the measure of u(Ω) and show that equality holds if and only if it is injective conformal. Also some sharp inequalities related to the L 2 norms of the radial derivatives of vector harmonic mappings from the unit ball in R n , n ≥ 2, are given. As an application, we estimate the 2-Dirichlet integrals of mappings in the Sobolev space W 2 1 .

Behavior of domain constants under conformal mappings

Israel Journal of Mathematics, 1995

Domain constants are numbers attached to regions in the complex plane C. For a region ~ in C, let d(f't) denote a generic domain constant. If there is an absolute constant M such that M-1 < d(f~)/d(A) <_ M whenever and A are conformally equivalent, then the domain constant is called quasiinvariant under conformal mappings. If M-1, the domain constant is conformally invariant. There are several standard problems to consider for domain constants. One is to obtain relationships among different domain constants. Another is to determine whether a given domain constant is conformally invariant or quasi-invariant. In the latter case one would like to determine the best bound for quasi-invariance. We also consider a third type of result. For certain domain constants we show there is an absolute constant N such that [d(f/)-d(A)[ < N whenever i2 and A are conformally equivalent, sometimes determining the best possible constant N. This distortion inequality is often stronger than quasi-invariance. We establish results of this type for six domain constants.

Conformal spectral stability estimates for the Dirichlet Laplacian

Mathematische Nachrichten

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains Ω ⊂ C using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc D. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of "conformal regular" domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such domains can have any Hausdorff dimension between one and two.

Sharp stability results for almost conformal maps in even dimensions

J Geom Anal, 1999

Let ~ C R n and n > 4 be even. We show that if a sequence {uJ } in W l'n/2 (f2; R n) is almost conformal in the sense that dist (Vu j , R+ SO(n) ) converges strongly to 0 in L n/2 and if u j converges weakly to u in W 1,n/2, then u is conformal and VuJ ~ Vu strongly in Lqoc for all 1 < q < n/2. It is known that this conclusion fails if n~2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f ( A ) that satisfies 0 < f ( A ) < C (1 + Iml n/2) and vanishes exactly on R + SO(n). The proof of these results involves the lwaniec-Martin characterization of conforrnal maps, the weak continuity and biting convergence of Jacobians, and the weak-L 1 estimates for Hodge decompositions.