Calabi-Yau manifolds with isolated conical singularities

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 Kähler-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 up to biholomorphism, the unique weak Ricci-flat Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler cone metric on \(\frac{1}{q}K_{Z}\) constructed using the Calabi ansatz. In particular, our result applies if \((X, \mathcal{O}(1))\) is a nodal quintic threefold in \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.     

Prescribing gaussian curvatures on surfaces with conical singularities

We consider prescribing Gaussian curvatures on surfaces with conical singularities. In a critical case, we obtain the best constant in an inequality. Then, by using the “distribution of mass” analysis, we are able to provide some sufficient conditions for a function to be the Gaussian curvature of some pointwise conformai metric.     

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

Complex powers of differential operators on manifolds with conical singularities

We construct the complex powersA _z for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatA _z exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofA _z , both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(A _z ). Supported by a Ford Foundation Fellowship administered by the National Research Council.     

Einstein metrics and complex singularities

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkähler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kähler metric (which is hyperkähler if and only if K_X is trivial), and that if K_X is strictly nef, then X also admits a complete (non-Kähler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number.Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable.All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.     

Metric homology for isolated conical singularities

The paper is devoted to the calculation of Local Metric Homology for spaces with isolated conical singularities. We show that in this case the result is similar to the corresponding result for Intersection Homology. The main tool is a parametric version of the following theorem of Federer and Fleming: small cycles are trivial.     

Bounded imaginary powers of differential operators on manifolds with conical singularities

 We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L _p -spaces over B, 1<p<∞. Under suitable ellipticity assumptions we can define a family of complex powers A ^z , zℂ. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations. Received: 12 June 2001; in final form: 3 June 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 35J70, 47A10, 35K57     

Electrostatics in a locally flat space with conical singularities

We calculate the potential of a pointlike source in a multicenter three-dimensional space-time and obtain general relations between the values of the regularized self-energy, force, and force moment. The self-action effects as well as the relative contribution of higher multipoles infinitely increase as the angle deficit increases. The results obtained are generalized to a system of parallel cosmic strings one of which carries a current. The case of string with a finite thickness is also considered. Translated from Teoreticheskaya i Matematicheskya Fizika, Vol. 123, No. 1, pp. 150–162, April, 2000.     

The fractional porous medium equation on manifolds with conical singularities II

This is the second of a series of two papers that studies the fractional porous medium equation, ${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma }{(|u|}^{m-1}u)=0$ with $m>0$ and $\sigma \in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_{\infty }-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.     

A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i -1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1~(**)(Σ, g)), we have■When b b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.     

Refined asymptotics for constant scalar curvature metrics with isolated singularities

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, , in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions. Oblatum 26-II-1997 & 6-II-1998 / Published online: 12 November 1998     

Relative critical exponents,non-vanishing and metrics with minimal singularities

In this article we establish a very general non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments combine the proof of the classical non-vanishing theorem due to V. Shokurov, as well as many ideas from Y.-T. Siu’s analytic proof of the finite generation of the canonical ring. An important tool is the notion of relative critical exponent of two closed positive currents with respect to a measure.     

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Kollár's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.