Related papers: A Calculus for Conformal Hypersurfaces and new hig…
We study numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The method we…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…
We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new…
We consider a compact pseudo-hermitian manifold (M,\theta, J), that is a manifold equipped with a contact form \theta and CR structure J. We consider a conformal deformation of the contact form to obtain a complete, singular contact form…
We introduce conformal anti-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a conformal submersion and find…
In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy…
We develop a calculus of variations for functionals on certain spaces of conformal maps. Such a space \Omega\ is composed of all maps that are conformal on domains containing a fix compact annular set of the Riemann sphere, and that are…
We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class $W^{2,q}$ with $q > 3$. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and…
The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains. In this paper, the conjugate function method is extended to cover conformal mappings between Riemannian…
We prove that the problem of constructing biharmonic conformal maps on a $4$-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition,…
The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds…
We study approximations to the Moreau envelope -- and infimal convolutions more broadly -- based on Laplace's method, a classical tool in analysis which ties certain integrals to suprema of their integrands. We believe the connection…
We report on the computation of invariants, covariants, and contravariants of cubic surfaces. All algorithms are implemented in the computer algebra system magma.
In this paper we deal with the global properties of Willmore surfaces in spheres via the harmonic conformal Gauss map using loop groups. We first derive a global description of those harmonic maps which can be realized as conformal Gauss…
Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a…
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a…
We derive a representation formula for harmonic polynomials and Laurent polynomials in terms of densities of the double-layer potential on bounded piecewise smooth and simply connected domains. From this result, we obtain a method for the…
In the realm of conformal geometry, we give a classification of the Euclidean hypersurfaces that admit a non-trivial conformal infinitesimal variation. In the restricted case of conformal variations, such a classification was obtained by E.…
We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincar\'e-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is…
We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We…