We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the "link" of the singular set. Within this class of "conic metrics," we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.
The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Euler characteristic $χ(Σ,β)$ is negative, then by a variational method, it is proved that there exists a constant $λ^\ast>0$ such that for any $λ\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_λ$; for any $λ$, $0<λ<λ^\ast$, there are at least two conformal…
We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2π$. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by …
In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than $2π$; in particular, we define and study the Teichmüller space $\mathcal{T}^{\mathrm{conic}}_{γ,k}$ of conic constant curvature metrics on a surface of genus $γ$ with $k$ conic points. The methods here are adopted from higher dimensional global analysis, generalizing Tromba's approach to the study of the standard Teichmüller space $\mathcal{T}_γ$. The main new ingredient is the theory of elliptic conic operators.
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2π$.
We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The key technique we used is to derive boundary C^2 estimates directly from C^0 estimates for fully nonlinear equations. The main result has appeared in the author's thesis.
A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $Σ$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $Σ\backslash \{{\rm singularities}\}$, called the {\it developing map} of the metric $g$. When the developing map $f$ of such a metric $g$ on the compact Riemann surface $Σ$ has reducible monodromy, we show that, up to some M{\" o}bius transformation on $f$, the logarithmic differential $d\,(\log\, f)$ of $f$ turns out to be an abelian differen…
A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.
We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This …
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization theorem for the sphere.
An HCMU metric is a conformal metric which has a finite number of singularities on a compact Riemann surface and satisfies the equation of the extremal Kähler metric. In this paper, we give a necessary and sufficient condition for the existence of a kind of HCMU metrics which has both cusp singularities and conical singularities.
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.
We extend to the conformal realm the concept of genuine deformations of submanifolds, introduced by Dajczer and the first author for the isometric case. Analogously to that case, we call a conformal deformation of a submanifold $M^n$ genuine if no open subset of $M^n$ can be included as a submanifold of a higher dimensional conformally deformable submanifold in such a way that the conformal deformation of the former is induced by a conformal deformation of the latter. We describe the geometric structure of a submanifold that admits a genuine conformal deformation and give several applications …
We consider conformal metrics of constant curvature 1 on a Riemann surface, with finitely many prescribed conic singularities and prescribed angles at these singularities. Especially interesting case which was studied by C. L. Chai, C. S Lin and C. L. Wang is described in some detail, with simplified proofs.
Let $X$ be a complex manifold and $S\hookrightarrow X$ be an embedding of complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle $N_{S}$ to a small neighborhood of $S$ in $X$ such that $F$ is in a precise sense holomorphic to the $(k-1)$-th order. Using this $F$ we obtain optimal estimates on asymptotical rates for asymptotically conical Calabi-Yau metrics constructed by Tian-Yau. Furthermore, when $S$ is …
In this paper we study Moebius applicable surfaces, i.e., conformally immersed surfaces in Moebius 3-space which admit deformations preserving the Moebius metric. We show new characterizations of Willmore surfaces, Bonnet surfaces and Harmonic inverse mean curvature surfaces in terms of Moebius or similarity invariants.
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 solutions of the system, which implies that a generic second order deformable surface is not isothermic. This differs from the situation in 3-dimensional conformal geometry, where isothermic surfaces coincide with deformable surfaces.
The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of critical Kähler Finsler metrics is obtained. A Yamabe type problem for mean Ricci curvature is considered.
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
In this paper, we study navigation problems on conic Kropina manifolds. Let $F(x, y)$ be a conic Kropina metric on an $n$-dimensional manifold $M$ and $V$ be a conformal vector field on $(M, F)$ with $F(x, - V_{x})\leq 1$. Let $\widetilde{F}= \widetilde{F} (x,y)$ be the solution of the navigation problem with navigation data $(F, V)$. We prove that $\widetilde{F}$ must be either a Randers metric or a Kropina metric. Then we establish the relationships between some curvature properties of $F$ and the corresponding properties of the new metric $\widetilde{F}$, which involve S-curvature, flag cur…
In this article, we found a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrödingier type operator. In particular, we proved a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigated compactly conformal deformations which either increase or decrease scalar curvature. We found local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist fo…
In this paper we classify Euclidean hypersurfaces $f\colon M^n \rightarrow \mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$. That $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$ is a genuine conformal deformation of $f$ means that it is a conformal immersion for which there exists no open subset $U \subset M^n$ such that the restriction $\tilde{f}|_U$ is a composition $\tilde f|_U=h\circ f|_U$ of $f|_U$ with a conformal immersion $h\colon V\to \mathbb{R}^{n+2}$ of an open subset $…
We discuss conformal deformation and warped products on some open manifolds. We discuss how these can be applied to construct Riemannian metrics with specific scalar curvature functions.
We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other hand, there is a one-to-one correspondence between infinitesimal conformal deformations in space and infinitesimal isometric deformations of the stereographic image on the sphere. The space of infinitesimal conformal deformations can be parametrized in terms of the change in dihed…
We study the deformation of spherical conical metrics with at least some of the cone angles larger than $2π$. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction considered in recent works of Mazzeo and author.
A spherical conical metric $g$ on a surface $Σ$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds $2π$. The eigenfunctions of the Friedrichs Laplacian $Δ_g$ with eigenvalue $λ=2$ play a special role in this problem, as they represent local obstructions to deformations of the metric $g$ in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eig…
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.
A simple proof is given of the necessary and sufficient condition on a triple of positive numbers A,B,C for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles A,B,C. The same condition is necessary and sufficient for the triple A,B,C to be interior angles of a spherical triangular membrane.
On pseudo-Riemannian manifolds of even dimension $n\geq 4$, with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signature the complex is elliptic.
We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the the…
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary.
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 $λ_1$ into the sphere is harmonic, establishing a link between conformal spectrum and har…
We study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of Cotton-York tensor and the L1-norm of it under the Ricci flow. In particular, we investigate the behavior of the L1-norm of the Cotton-York tensor under the Ricci flow on three-dimensional simply-connected homogeneous manifolds which admit compact quotients. For a non-homogeneous case, we also investigate the behavior of the L1-norm of the Cotton-York tensor for the product metric of the Rosenau solution for the Ricci flow on a two-sphere and the standard metric of a cir…
We obtain a structure theorem for the group of holomorphic automorphisms of a conformally Kähler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the Kähler-Einstein, cscK, and extremal Kähler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally Kähler, Einstein--Maxwell metrics on $\mathbb{CP}^1 \times \mathbb{CP}^1$. We also use our result in order to introduce a (relative) Mabuchi energy in the more general context of $(K, q, a)$-extremal Kähler metrics in a …
In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. Secondly, we prove that the formal deformation problem is governed by an $L_\infty$-structure which is a $\frak b$-deformation of strong homotopy Lie algebroids introduced in Oh and Park (2005) in the symplectic context. Then we study deformations of locally conformal symplectic…
In \cite{Luo0}, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant curvature PL-metric.
The aim of this paper is to develop on the 1-jet space J^1(R, M^n) the Finsler-like geometry (in the sense of distinguished (d-) connection, d-torsions, d-curvatures and some gravitational-like and electromagnetic-like geometrical models) attached to the (t,x)-conformal deformation of the Berwald-Moor metric.
We describe the range of the Radon transform on the space $M$ of irreducible conics in $\CP^2$ in terms of natural differential operators associated to the $SO(3)$-structure on $M=SL(3, \R)/SO(3)$ and its complexification. Following \cite{moraru} we show that for any function $F$ in this range, the zero locus of $F$ is a four-manifold admitting an anti-self-dual conformal structure which contains three different scalar-flat Kähler metrics. The corresponding twistor space ${\mathcal Z}$ admits a holomorphic fibration over $\CP^2$. In the special case where ${\mathcal Z}=\CP^3\setminus\CP^1$ the…
J. Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one. We prove by developing map that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4α^2\vert z \vert^{2α-2}}{(1-\vert z \vert ^{2α})^2}\vert \mathrm{d} z \vert^2}$ with $α>0$ or $\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|dz|^{2}}$.
The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the obstruction tensor. This new definition of the obstruction tensor lea…
Registration, which aims to find an optimal one-to-one correspondence between different data, is an important problem in various fields. This problem is especially challenging when large deformations occur. In this paper, we present a novel algorithm to obtain diffeomorphic image or surface registrations with large deformations via quasi-conformal maps. The basic idea is to minimize an energy functional involving a Beltrami coefficient term, which measures the distortion of the quasi-conformal map. The Beltrami coefficient effectively controls the bijectivity and smoothness of the registration…
