Searcharxiv⌕ Search

arXiv subjects

F. Reese Harvey

Publications and source records attributed to F. Reese Harvey.

At least 37 records · Page 2Linked to original sources
arXiv2014

Smooth Approximation of Plurisubharmonic Functions on Almost Complex Manifolds

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.

math.CV↗
arXiv2014

Tangents to subsolutions -- existence and uniqueness, Part II

This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in ${\mathbb R}^n$. An upper semi-continuous function u on an open set $Ω$ in ${\mathbb R}^n$ is G-plurisubharmonic if its restriction to $Ω\cap W$ is subharmonic for every affine G-plane $W$. Here G is assumed to be invariant under a subgroup K of O(n) which acts transitively on the sphere $S^{n-1}$. Tangents to u at a point x are the cluster points of u under a natural flow (or blow-up) at x. They always exist and are G-harmonic at all points of continuity. A homogeneity property is established for all tangents in these geometric cases. This leads to principal results concerning the Strong Uniqueness of Tangents, which means that all tangents are unique and of the form $ΘK_p$ where $K_p$ is the Riesz kernel and $Θ$ is the density of u at the point. Strong uniqueness is a form of regularity which implies that the sets $\{Θ(u,x)\geq c\}$ for $c>0$ are discrete. When the invariance group K= O(n), U(n) or Sp(n), strong uniqueness holds for all but a small handful of cases. It also holds for essentially all interesting G which arise in calibrated geometry. When strong uniqueness fails, homogeneity implies that tangents are characterized by a subequation on the sphere, which is worked out in detail. In the cases corresponding to the real, complex and quaternionic Monge-Ampère equations (convex functions, and complex and quaternionic plurisubharmonic functions) tangents, which are far from unique, are then systematically studied and classified.

math.AP↗
arXiv2014

Removable singularities for nonlinear subequations

We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E in X have the property that every F-subharmonic function (subsolution) on X-E, which is locally bounded across E, extends to an F-subharmonic function on X. We also study the related question for F-harmonic functions (solutions) which are continuous across E. Main results assert that if there exists a convex cone subequation M such that F+M is contained in F, then any closed set E which is M-polar has these properties. To be M-polar means that E = {f = -\infty} where f is M-subharmonic on X and smooth outside of E. Many examples and generalizations are given. These include removable singularity results for all branches of the complex and quaternionic Monge-Ampere equations, and a general removable singularity result for the harmonics of geometrically defined subequations. For pure second-order subequations in R^n with monotonicity cone M, the Riesz characteristic p = p(M) is introduced, and extension theorems are proved for any closed singular set E of locally finite Hausdorff (p-2)-measure (or, more generally, of (p-2)-capacity zero). This applies for example to branches of the equation s_k(D^2 u) = 0 (kth elementary function) where p(M) = n/k. For convex cone subequations themselves, several removable singularity theorems are proved, independent of the results above.

math.AP↗
arXiv2013

p-convexity, p-plurisubharmonicity and the Levi problem

Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables, namely: local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the p-hull of the boundary union with the "core" of the space. Lastly, the exteme rays in the convex cone of p-positive matrices are characterized. This is a basic result with many applications.

math.DG↗
arXiv2013

Potential Theory on Almost Complex Manifolds

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. In the second theorem, using our "viscosity" definitions, the Dirichlet problem is solved for the complex Monge-Ampère equation in both the homogeneous and inhomogeneous forms. These two results are based on theorems found in two recent papers of the authors. Finally, it is shown that the plurisubharmonic functions considered here agree with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.

math.CV↗
arXiv2013

The Restriction Theorem for Fully Nonlinear Subequations

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy the same constraints. We first prove an elementary result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any "geometrically defined" subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.

math.AP↗
arXiv2013

The equivalence of viscosity and distributional subsolutions for convex subequations - a strong Bellman principle

There are two useful ways to extend nonlinear partial differential inequalities of second order: one uses viscosity theory and the other uses the theory of distributions. This paper considers the convex situation where both extensions can be applied. The main result is that under a natural "second-order completeness" hypothesis, the two sets of extensions are isomorphic, in a sense that is made precise.

math.AP↗
arXiv2011

Geometric plurisubharmonicity and convexity - an introduction

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). A surprising number of results in complex analysis carry over to this more general setting. The notions of: a G-submanifold, an upper semi-continuous G-plurisubharmonic function, a G-convex domain, a G-harmonic function, and a G-free submanifold, are defined. Results include a restriction theorem as well as the existence and uniqueness of solutions to the Dirichlet Problem for G-harmonic functions on G-convex domains.

math.DG↗
arXiv2011

Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Ampere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain D in X, we give geometric conditions which imply that the Dirichlet problem on D is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then associate to F two natural "conical subequations": a monotonicity subequation M and the asymptotic interior of F. If X carries a global M-subharmonic function, then weak comparison implies full comparison. The asymptotic interior of F is used to formulate boundary convexity and provide barriers. In combination the Dirichlet problem becomes uniquely solvable as claimed. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.

math.AP↗
arXiv2010

Split Special Lagrangian Geometry

One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is replaced by tau with tau^2 = 1. A rather surprising amount of complex geometry carries over, almost untouched, and this has been the subject of many papers. We briefly review this material and, in particular, we discuss Hermitian D-manifolds with trivial canonical bundle, which provide the background space for the geometry of split SLAG submanifolds. A removable singularities result is proved for split SLAG subvarieties. It implies, in particular, that there exist no split SLAG cones, smooth outside the origin, other than planes. This is in sharp contrast to the complex case. Parallel to the complex case, space-like Lagrangian submanifolds are stationary if and only if they are theta-split SLAG for some phase angle theta, and infinitesimal deformations of split SLAG submanifolds are characterized by harmonic 1-forms on the submanifold. We also briefly review the recent work of Kim, McCann and Warren who have shown that split Special Lagrangian geometry is directly related to the Monge-Kantorovich mass transport problem.

math.DG↗
arXiv2010

Hyperbolic polynomials and the Dirichlet problem

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. In a second, independent part of the paper, the relationship of Garding's theory to the authors' recent work (arXiv:0710.3991) on the Dirichlet problem for fully nonlinear partial differential equations is investigated. Let p be a homogeneous polynomial of degree m on S^2(R^n) which is hyperbolic with respect to the all positive directions A \geq 0. Then p has an associated eigenvalue map lambda:S^2(R^n) \to R^m, defined modulo the permutation group acting on R^m. Consequently, each closed symmetric set E of R^m induces a second-order p.d.e. by requiring, for a C^2-function u in n-variables, that (D^2 u)(x) lie in the boundary of E for all x. Assume that E + (R_+)^m is contained in E. A main result is that for smooth domains in R^n whose boundary is suitably (p,E)-pseudo-convex, the Dirichlet problem has a unique continuous solution for all continuous boundary data. This applies to a vast collection of examples the most basic of which are the m distinct branches of the equation p(D^2 u) =0. In the authors' recent extension of results from euclidean domains to domains in riemannian manifolds (arXiv:0907.1981), a new global ingredient, called a monotonicity subequation, was introduced. It is shown in this paper that for every polynomial $p$ as above, the associated Garding cone is a monotonicity cone for all branches of the the equation p(Hess u) = 0 where Hess u denotes the riemannian Hessian of u.

math.AP↗
arXiv2009

An Introduction to Potential Theory in Calibrated Geometry

We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. In particular, the notion of pseudo-convexity for a calibrated manifold (X,ϕ) is introduced and studied. Analogues of totally real submanifolds are also introduced and used to construct enormous families of strictly ϕ-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout.

math.DG↗
arXiv2009

Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,ϕ). In particular, on X there exist ϕ-plurisubharmonic functions, ϕ-convex domains, ϕ-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the ϕ-submanifolds, or more generally, the ϕ-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the ϕ-convex hull of a compact set K in X in terms of ϕ-positive Green's currents on X and Jensen measures on K. We also characterize boundaries of ϕ-currents entirely in terms of ϕ-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.

math.DG↗
arXiv2008

Plurisubharmonicity in a General Geometric Context

Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamonicity and convexity. It also applies in a number of non-geometric situations. Results include: fundamental properties of $P^+$-plurisubharmonic functions, plurisubharmonic distributions and regularity, $P^+$-convex domains and $P^+$-convex boundaries, topological restrictions on and construction of such domains, continuity of upper envelopes, and solutions of the Dirichlet problem for related Monge-Ampere-type equations. Many results in this paper have been generalized in recent work of the authors. However, this article covers many cases of geometric interest, and certain convexity assumptions here allow the use of classical analytic methods, making the exposition more accessible.

math.DG↗
arXiv2007

Dirichlet Duality and the Nonlinear Dirichlet Problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric ``F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of the Special Lagrangian potential equation.

math.AP↗
arXiv2007

Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part I

We introduce the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z in P^n - M of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it generalizes directly from P^n to any projective manifold. Part 1 of this paper establishes the following result for the case p=1. Let M be an oriented, stable, real analytic curve in P^n. Then M is the boundary of a positive holomorphic 1-chain T with Mass(T) < K in P^n if and only if Link(M,Z) > -K deg(Z) for all algebraic hypersurfaces Z in P^n - M. An analogous theorem is implied in any projective manifold. Part 2 of this paper studies similar results for p>1.

math.CV↗
arXiv2007

The Projective Hull of Certain Curves in C^2

The projective hull X^ of a subset X in complex projective space P^n is an analogue of the classical polynomial hull of a set in C^n. If X is contained in an affine chart C^n on P^n, then the affine part of X^ is the set of points x in C^n for which there exists a constant M=M_x so that |p(x)| < M^d sup{|p(y)| : y in X} for all polynomials p of degree less than or equal to d, and any d > 0. Let X^(M) be the set of points x where M_x can be chosen < M. Using an argument of E. Bishop, we show the following. Let G be a compact real analytic curve (not necessarily connected) in C^2. Then for any linear projection p: C^2 --> C^1, the set of points in G^(M) lying above a point z in C^1 is finite for almost all z. Using this, we prove the conjecture that for any compact stable real-analytic curve G in P^n, the set G^-G is a 1-dimensional complex analytic subvariety of P^n-G.

math.CV↗