Searcharxiv⌕ Search

arXiv subjects

Yanir A. Rubinstein

Publications and source records attributed to Yanir A. Rubinstein.

52 records · Page 3Linked to original sources
arXiv2014

On discontinuity of planar optimal transport maps

Consider two bounded domains $Ω$ and $Λ$ in $\mathbb{R}^{2}$, and two sufficiently regular probability measures $μ$ and $ν$ supported on them. By Brenier's theorem, there exists a unique transportation map $T$ satisfying $T_\#μ=ν$ and minimizing the quadratic cost $\int_{\mathbb{R}^{n}}|T(x)-x|^{2}dμ(x)$. Furthermore, by Caffarelli's regularity theory for the real Monge--Ampère equations, if $Λ$ is convex, $T$ is continuous. We study the reverse problem, namely, when is $T$ discontinuous if $Λ$ fails to be convex? We prove a result guaranteeing the discontinuity of $T$ in terms of the geometries of $Λ$ and $Ω$ in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of $\partialΛ$ to distinguish between Brenier and Alexandrov weak solutions of the Monge--Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

math.AP↗
arXiv2014

Smooth and singular Kahler-Einstein metrics

Smooth Kahler-Einstein metrics have been studied for the past 80 years. More recently, singular Kahler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a survey of some of these developments.

math.DG↗
arXiv2014

Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere equation. Second, we prove the second order regularity of the solution of the free-boundary problem for the Laplace equation with a rooftop obstacle, based on a new a priori estimate on the size of balls that lie above the non-contact set. As an application, we prove that convex- and plurisubharmonic-envelopes of rooftop obstacles have bounded second derivatives.

math.AP↗
arXiv2013

Analysis of polarity

We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This analysis may be used to solve a family of first order equations reminiscent of Hamilton--Jacobi and conservation law equations, as well as some second order Monge-Ampere type equations. A special case of the latter, that we refer to as the homogeneous polar Monge--Ampere equation, gives rise to a canonical method of interpolating between convex functions.

math.AP↗
arXiv2013

Asymptotically log Fano varieties

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth log Fano varieties. Based on this classification we formulate an asymptotic logarithmic version of Calabi's conjecture for del Pezzo surfaces for the existence of Kähler--Einstein edge metrics in this regime. We make some initial progress towards its proof by demonstrating some existence and non-existence results, among them a generalization of Matsushima's result on the reductivity of the automorphism group of the pair, and results on log canonical thresholds of pairs. One by-product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime, of which a folklore conjecture concerning the case of a Fano manifold with an anticanonical divisor is a special case.

math.AG↗
arXiv2012

The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan

We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C^3 lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C^3, in the sense that there exists a dense set of C^3 Cauchy data for which there exists no C^3 solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to C^1 solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak C^1 solution afterwards. Finally, we introduce the notion of a "leafwise subsolution" for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object.

math.DG↗
arXiv2011

Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, the geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.

math.DG↗
arXiv2011

Ricci flow and the metric completion of the space of Kahler metrics

We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kahler metrics, making contact with recent generalizations of the Calabi-Yau Theorem due to Dinew, Guedj-Zeriahi, and Kolodziej. As an application, we obtain a new analytic stability criterion for the existence of a Kahler-Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kahler-Ricci flow converges as soon as it converges in the metric sense.

math.DG↗
arXiv2010

The Cauchy problem for the homogeneous Monge-Ampere equation, I. Toeplitz quantization

The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) this "quantum analytic continuation potential" coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel we prove that the Legendre transform potential ceases to solve the HRMA after that time.

math.DG↗
arXiv2010

The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article---that uses tools of convex analysis and can be read independently---we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable.

math.AP↗
arXiv2008

Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product of the manifold with a Riemann surface with S^1 boundary admit such approximations. We also show that the Eells-Sampson flow on the space of Kahler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform.

math.DG↗
arXiv2007

The Ricci iteration and its applications

In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kahler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kahler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves.

math.DG↗
arXiv2007

On energy functionals, Kahler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

We prove that the existence of a Kahler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kahler potentials on which this inequality holds on higher-dimensional Fano Kahler-Einstein manifolds.

math.DG↗
arXiv2007

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I

In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kahler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kahler and conformal geometry such as constant scalar curvature metrics, Kahler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kahler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.

math.DG↗