A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Exterior Monge-Ampère solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R².     

The Bieberbach-Rademacher problem for the Monge-Ampère operator

Assume that Ω is a bounded, strictly convex, smooth domain in ℝ^N withN≥2. We consider the problem det ((∂ _iju(x)))=f(x,u(x)),u(x)→∞ asx→∂Ω, where (∂ _iju(x)) denotes the Hessian ofu(x) andf meets some natural regularity and growth conditions. We prove that there exists a unique smooth, strictly convex solution of this problem. The boundary-blow-up rate ofu(x) is characterized in terms of the distance ofx from ∂Ω. Partially supported by the Royal Swedish Academy of Sciences, Gustaf Sigurd Magnuson's fund.     

The Dirichlet heat kernel in inner uniform domains: Local results,compact domains and non-symmetric forms

This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable real coefficients in inner uniform domains in Rⁿ${\mathbb{R}}^n$. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.     

On the nondegeneracy of the critical points of the Robin function in symmetric domainsSur le non-dégénérescence des points critiques de la fonction de Robin dans les domaines symétriques

Let $\text{Ω}$ be a smooth bounded domain of $\text{R}{}^{\mathrm{N}}$, N?2, which is symmetric with respect to the origin. In this Note we prove that, under some geometrical condition on $\text{Ω}$ (for example convexity in the directions x₁,…,x_N), the Hessian matrix of the Robin function computed at zero is diagonal and strictly negative definite. To cite this article: M. Grossi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 157–160.     

Concentration of measures supported on the cube

We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube [0, 1] ⁿ ? ? ⁿ whose density takes the form exp(?ψ), where the function ψ is assumed to be convex (but not strictly convex) with bounded pure second derivatives. Our argument relies on a transportation-cost inequality á la Talagrand.     

Proper holomorphic maps between Reinhardt domains and strictly pseudoconvex domains

LetR be a Reinhardt domain andD a bounded simply connected strictly pseudoconvex domain withC ^∞ boundary. We prove that any proper holomorphic mapF:R →D is, up to biholomorphism ofD, of the form \((z_1^{d_1 } , z_2^{d_2 } , \ldots , z_n^{d_n } )\) withd ₁,d ₂,…,d _n ∈ IN.     

Automorphisms of hyperbolic domains inR n

The main result of this paper is a fixed-point theorem for projective automorphisms of a bounded strongly convex domain inR ⁿ . Several corollaries and applications are derived, especially on the dimension of the full automorphism group in the smooth case.     

A priori estimate of\parallel u\parallel _{c^2 (\overline \Omega )}for convex solutions of the Dirichlet problem for the Monge-Ampere equations

One gives an a priori estimate of \(\parallel u\parallel _{c^2 (\overline \Omega )}\) for the convex solutions of the Dirichlet problem for the Monge-Ampère equation (U_ij=f(x,u,u_x)in a strictly convex bounded domain det,ΩcRⁿ     

Diameters of sections and coverings of convex bodies

We study the diameters of sections of convex bodies in R^N determined by a random N×n matrix Γ, either as kernels of Γ^* or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in R^N has one well bounded k-codimensional section, then for any m>ck random sections of K of codimension m are also well bounded, where c?1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c=1.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

On a boundary analog of the Forelli theorem

A boundary analog of the Forelli theorem for real-analytic functions is established, i.e., it is demonstrated that each real-analytic function f defined on the boundary of a bounded strictly convex domain D in the multidimensional complex space with the one-dimensional holomorphic extension property along families of complex lines passing through a boundary point and intersecting D admits a holomorphic extension to D as a function of many complex variables.     

Monge-Ampère equations and moduli spaces of manifolds of circular type

A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point xo, so that: (a) it is a smooth solution on M?{xo} to the Monge-Ampère equation n(ddcu)=0; (b) xo is a singular point for u of logarithmic type and eu extends smoothly on the blow up of M at xo; (c) ddc(eu)>0 at any point of M?{xo}. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of Cn.A set of modular parameters for bounded manifolds of circular type is considered. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n² when M is a (non-convex) circular domain while it is of dimension n²+2n when M is a strictly linearly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.     

The robust minimum spanning tree problem: Compact and convex uncertainty

We consider the robust minimum spanning tree problem where edges costs are on a compact and convex subset of Rⁿ. We give the location of the robust deviation scenarios for a tree and characterizations of strictly strong edges and non-weak edges leading to recognition algorithms.     

On Minimal Annuli with a Straight Line Boundary

Shiffman proved that if a minimal annulus A in a slab is bounded by two convex Jordan curves contained respectively in the two boundary planes P and Q of the slab, then A intersects all parallel planes between P and Q in strictly convex curves. We generalize Shiffman's result to the case that A is bounded by a strictly convex C² Jordan curve and a straight line. We show that in this case Shiffman's result is still true.     

Approximation of linear force-free fields in bounded 3-D domains

We introduce and study some methods and algorithms for approximating linear forcefree fields solution of the equation curl B = αB in a bounded domain of R³. Several numerical tests are done in order to study approximation error, compare algorithms, and display some additional properties of force-free fields.     

Exit time and invariant measure asymptotics for small noise constrained diffusions

Constrained diffusions, with diffusion matrix scaled by small ?>0, in a convex polyhedral cone G⊂R^k, are considered. Under suitable stability assumptions small noise asymptotic properties of invariant measures and exit times from domains are studied. Let B⊂G be a bounded domain. Under conditions, an “exponential leveling” property that says that, as ?→0, the moments of functionals of exit location from B, corresponding to distinct initial conditions, coalesce asymptotically at an exponential rate, is established. It is shown that, with appropriate conditions, difference of moments of a typical exit time functional with a sub-logarithmic growth, for distinct initial conditions in suitable compact subsets of B, is asymptotically bounded. Furthermore, as initial conditions approach 0 at a rate ?² these moments are shown to asymptotically coalesce at an exponential rate.     

A logarithmic-quadratic proximal point scalarization method for multiobjective programming

We present a proximal point method to solve multiobjective programming problems based on the scalarization for maps. We build a family of convex scalar strict representations of a convex map F from R ⁿ   to  R ^m with respect to the lexicographic order on R ^m and we add a variant of the logarithmic-quadratic regularization of Auslender, where the unconstrained variables in the domain of F are introduced in the quadratic term. The nonegative variables employed in the scalarization are placed in the logarithmic term. We show that the central trajectory of the scalarized problem is bounded and converges to a weak pareto solution of the multiobjective optimization problem.     

Reduction of variance for Gaussian densities via restriction to convex sets

Let X be a random vector with values in $\text{R}$ⁿ and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on $\text{R}$ⁿ. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ $\text{R}$ⁿ. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.     

A global minimization algorithm for a class of one-dimensional functions

An algorithm is developed for finding the global minimum of a continuously differentiable function on a compact interval in R₁. The function is assumed to be the sum of a convex and a concave function, each of which belongs to C¹[a, b]. Any one-dimensional function with a bounded second derivative can be so written and, therefore, such functions generally have many local minima. The algorithm utilizes the structure of the objective to produce an ?-optimal solution by a sequence of simple one-dimensional convex programs.