arXiv research

Paper finds singular solutions for a specific physics problem on a sphere.

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…

Study on Laplacian flow and coflow of Locally Conformal Parallel G2-structures.

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold …

Study BGG operators on homogeneous conformal geometries.

We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…

BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of …

We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equatio…

The paper proves uniqueness of a solution in general relativity.

We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the geometry of the center of mass of such solutions. We also review methods to con…

The round sphere is the only solution to inverse curvature flows under certain conditions.

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

The study explores conformal planes with finite areas.

The paper solves fractional scalar curvature problems on conformal infinities.

Characterizes Kerr spacetimes using conformal methods.

We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in ${\bf R}^n$. We consider the case when the scalar curvature of the conformal metric is bounded between two positive numbers outside a compact set. It is shown that the solution has slow decay if the radial change is …

Study on solutions to complex equations, proving strong comparison and Liouville theorems.

Paper studies solutions to a specific equation in conformal geometry with singular sets.

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic probl…

Singular Yamabe problems involve changing sign solutions with interesting geometric properties.

Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifo…

Study improves understanding of solutions to complex equations in geometry.

Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.

New method approximates Gaussian curvature on discrete surfaces.

Paper proves existence of solutions for a specific system.

Sharp rates and symmetry for higher order conformally invariant equations near singularities.

New equations define frustrated conformal transformations generating maps on ambitwistor space.

Study on solutions to conformally invariant fourth order equations, classifying their properties.

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existen…

We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and singular solutions. In particular, we show that isothermic surfaces are singular sol…

We construct infinite-dimensional families of non-singular static space times, solutions of the vacuum Einstein-Maxwell equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity.

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant me…

The conformal method has been effective for parametrizing solutions to the Einstein constraint equations on closed 3-manifolds. However, it is still not well-understood; for example, existence of solutions to the conformal equations for zero or negative Yamabe metrics is still unknown without the so-called ``CMC'' or `…

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe metrics and with TT-tensor $σ$ = 0 have a non-trivial solution, and thus answer a que…

We study solutions to conformally invariant equations with isolated singularties.

The paper explores projective structures on curves and their applications in conformal geometry.

Classification of ground state solutions to critical Dirac equation on spheres.

Study on solutions near isolated singularities in 6D Yamabe equation.

Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally Kähler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-CP_2).The same method also yields analogous sol…

Study concavity of solutions to elliptic equations under conformal deformations.

The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal…

The paper develops a comprehensive theory of submanifolds in conformal geometries.

We define a generalization of convex functions, which we call $δ$-convex functions, and show they must satisfy interior Hölder and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative …

New heat flow for harmonic maps avoids singularities but not bubbles.

The constraint equations of general relativity can in many cases be solved by the conformal method. We show that a slight modification of the equations of the conformal method admits no solution for a broad range of parameters. This suggests that the question of existence or non-existence of solutions to the original e…

Generalising the results in arXiv:1612.00281, we construct infinite-dimensional families of non-singular stationary space times, solutions of Yang-Mills-Higgs-Einstein-Maxwell-Chern-Simons-dilaton-scalar field equations with a negative cosmological constant. The families include an infinite-dimensional family of soluti…