We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreover we give formulae related to remote singularities of the boundary-envelope.
This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
We give global restrictions on the possible boundaries of compact, orientable, locally conformally flat manifolds of dimension $4k$ in terms of integrality of eta invariants.
Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we find all of the local models for an isolated singularity of a flat metric whose area satisfies some polynomial growth condition near the singularity. In particular, we show that an isolated singularity of a flat metric with finite area is also a conical one.
Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classification of the ALE Ricci-flat Kahler surfaces. We construct ALF Ricci-flat Kahler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat Kahler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.
It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,ν)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this pape…
It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we shall prove that generically flat fronts in the hyperbolic 3-space admit only cuspidal edges and swallowtails. Moreover, we will show that every complete flat front (which is not rotationally symmetric) has associated …
By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$. By a result of W.~Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a…
We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the proof we analyze the decay rates of solutions to the Bach-flat equation linearized around a flat metric. This classification is used to prove that Bach-flat cones are in fact ALE of order $τ$ for any $τ< 2$. This result is then used to prove the removal of singularities theorem.
We construct Ricci flat Kahler metrics with cone singularities along a complex hypersurface. This construction is inspired in part by R. Mazzeo's program in the case of negative Einstein constant, and uses the linear theory developed recently by S. Donaldson.
We study the shortest geodesics on flat cone spheres, i.e. flat metrics on the sphere with conical singularities. The length of the shortest geodesic between two singular points can be treated as a function on the moduli space of flat cone spheres with prescribed angle defects. We prove a recurrent relation on the distribution of this function with respect to Thurston's volume form on the moduli space.
We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study the blow up behavior of $g(t)$ at the maximal time of existence, $T < \infty$. We assume that $(M,g(\cdot, t))$ satisfies (i) the injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time $t = 0$ and the scalar curvature bounded at each time-slice. We show tha…
The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by M. Gromov in 1983, who proposed an argument toward the existence of $L^2$-extremizers exploiting the theory of $r$-regularity developed by P. A. White and others by the 1950s. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures…
We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the di…
We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities as log del Pezzo surfaces with only cyclic quotient singularities at infinity.
On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics on elliptic K3 surfaces. We construct many more such metrics on S^2, singular at any 6 or more points, and compute the local affine structure near the singularities. The techniques involve affine differential geometry and solving a semilinear PDE on S^2 minus singularities. We a…
We extend our discrete uniformization theorems for planar, $m$-connected, Jordan domains [Journal für die reine und angewandte Mathematik 670 (2012), 65--92] to closed surfaces of non-positive genus.
We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which have higher co-dimension and the fact that using the flat norm to compute distances not only gives a distance, but also an informative decomposition of the distance. This decomposition is made to depend on scale using the "flat norm with scale" which we define in direct analogy to …
Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic boundaries of Levi-flat hypersurfaces in terms of their normal coordinates. For the remaining case of generic real analytic boundary we get a weaker extension theorem. We find examples to show that these two extension results are optimal. Further, a class of nowhere minimal real ana…
We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be "approached linearly". The homeomorphism type of L(x) is an affine invariant. When x is not a cone point or an infinite-angle singularity, we say it is wild; in this case it is necessary to add further metric data to L(x) to get a quantitative description of the surface near x.
We give a characterisation of Bieberbach manifolds which are geodesic boundaries of a compact flat manifold, and discuss the low dimensional cases, up to dimension 4.
Consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$.
We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. We also examine discrete linear Weingarten surfaces of Bryant type in hyperbolic 3-space, and consider an example of a discrete flat surface related to the Airy equation that exhibits…
There are three main components to this article: (i) A formula for the eta invariant of the signature complex for any finite subgroup of ${\rm{SO}}(4)$ acting freely on $S^3$ is given. An application of this is a non-existence result for Ricci-flat ALE metrics on certain spaces. (ii) A formula for the orbifold correction term that arises in the index of the self-dual deformation complex is proved for all finite subgroups of ${\rm{SO}}(4)$ which act freely on $S^3$. Some applications of this formula to the realm of self-dual and scalar-flat Kähler metrics are also discussed. (iii) Two infinite …
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.
We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded. Moreover, we shall give new examples for which equality holds.
We consider a class of globally hyperbolic space-times with "expanding singularities". Under suitable assumptions we show that no $C^0$-extensions across a compact boundary exist, while the boundary must be null wherever differentiable (which is almost everywhere) in the non-compact case.
In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a special type of a (possibly immersed) genus $(m-1)$ singular flat surface, tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$. In part I the solution of a Dirichlet problem defined o…
We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes can only jump down at countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Xu and the author, we show that K-semistability is a very generic or empty condition in any $\mathbb{Q}$-Gorenstein flat family of log Fano pairs.
We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for c…
We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$.
We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2+1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse their geometrical properties, introduce a generalised notion of global hyperbolicity and classify all stationary flat spacetimes with singularities that are globally hyperbolic in that sense. We then apply our results to (2+1)-gravity and analyse the causality structure of these s…
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.
In this paper we derive a refined asymptotic expansion, near an isolated singularity, for conformally flat metrics with constant positive Q-curvature and positive scalar curvature. The condition that the metric has constant Q-curvature forces the conformal factor to satisfy a fourth order nonlinear partial differential equation with critical Sobolev growth, whose leading term is the bilaplacian. We model our results on a similar asymptotic expansion for conformally flat, constant scalar curvature metrics proven by Korevaar, Mazzeo, Pacard, and Schoen. Along the way we analyze the linearization…
In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with the harmonic measure. As an application of these results we prove that the geodesic flow on a non-flat finite volume harmonic manifold without conjugate points is topologically transitive.
Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point $x \in X$ there exist a Kahler-Einstein Fano manifold $Z$ and a positive integer $q$ dividing $K_Z$ such that $-\frac{1}{q}K_Z$ is very ample and such that the germ $(X,x)$ is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of $\frac{1}{q}K_{Z}$. We prove that …
We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case are: a) different ``interaction energies'' between pairs of conical singularities than one would expect from a naive extrapolation of the results for a smooth surface; and b) ``self-energies'' of the singularities.
We analyze the relevance of the generalized Kronheimer construction for the gauge-gravity correspondence. We study the general structure of IIB supergravity D3-brane solutions on crepant resolutions $Y$ of singularities $\mathbb{C}^3/Γ$ with $Γ$ a finite subgroup of $SU(3)$. Next we concentrate on another essential item for the D3-brane construction, i.e., the existence of a Ricci-flat metric on $Y$, with particular attention to the case $Γ=\mathbb{Z}_4$. We conjecture that on the exceptional divisor the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincid…
