In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.
We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.
Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these boundary value problems are elliptic and self--adjoint. We show that under certain periodic deformations of the boundary data our operators exhibit non-trivial spectral flow.
We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elementary prove that E has compact leaves if and only if H is elliptic. In this case the leaves of E give explicitly the elliptic fibration of H, and the natural orbifold structure on the leaf space is illustrated.
We study the problem of conformally deforming a manifold with boundary to have vanishing σ4-curvature in the interior and constant H4- curvature on the boundary. We prove that there are geometrically distinct solutions using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of a sphere and hyperbolic space only works for a finite set of dimensions.
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.
Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show that, when the dimension is greater than or equal to 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.
In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by $C^{1,1}$ solutions on larger and larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$ solutions: they are either constants or standard bubbles.
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.
We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.
We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We show that Yamabe constant is monotonically increasing along conformal Ricci flow on compact manifolds. We also show that conformal Ricci flow is the gradient flow for the ADM mass on asymptotically flat manifolds.
We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fixed conformally congruent area. On the way to establish this result we prove that tori which are critical points of the area for perturbations within a given conformal class and which are degenerate points of the conformal class mapping - i.e. isothermic - are either minimal surf…
In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a priori estimates combined with the perturbation result of Part I allow us to obtain some existence results. On the six dimensional sphere we prove the existence of at least one solution when an associated index is different from zero.
We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with $g_0$ the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, Delta K_g + e^(2u)=0, with x in R^2, describing a conformally flat surface with a Gauss curvature function K_g that is generated self-consistently through the metric's conformal factor.
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE . It enable us also to establish new results. In particular we solve a conjecture by E.Heinz asserting…
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 obtain local pointwise second derivative estimates for $W^{2,p}$-strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry.
We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.
We study conformal harmonic coordinates on Riemannian manifolds. These are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show their existence under general conditions. We find that conformal harmonic coordinates are a close conformal analogue of harmonic coordinates. We prove up to boundary regularity results for conformal mappings. We show that Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors become elliptic operators in conformal harmonic coordinates if one normalizes the determinant of the metric. We give a corresponding elliptic regular…
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven by Domokos & Manfredi (Cont. Math. 2005). In turn, the a priori estimates found are shown to imply the suitable loca…
We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We further obtain sharp, quantitative regularity estimates for non-negative limiting solutions.
We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new proof of an estimate of Streets-Warren on the twisted real Monge-Ampere equation.
In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge--Ampère equation for $(n-1)$-plurisubharmonic equations in the almost Hermitian setting.
We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.
We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.
A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.
We derive a priori $C^2$ estimates for a class of complex Monge-Ampere type equations on Hermitian manifolds. As an application we solve the Dirichlet problem for these equations under the assumption of existence of a subsolution; the existence result, as well as the second order boundary estimates, is new even for bounded domains in $\bfC^n$.
