On the Gaussian Curvature of Graphs of Holomorphic Functions

We show that the graph\[Γf={(z,f(z))∈C2:z∈S}\Gamma _f=\{(z,f(z))\in {\mathbb {C}}^2:\,z\in S\}\]inC2{\mathbb {C}}^2of a functionffon the unit circleSSwhich is either continuous and quasianalytic in the sense of Bernstein orC∞C^\inftyand quasianalytic in the sense of Denjoy is pluripolar.

We study some properties of graphs whose mean curvature (in distributional sense) is a vector Radon measure. In particular, we prove that the distributional mean curvature of the graph of a Lipschitz continuous function u is a measure if and only if the distributional divergence of T u is a measure. This equivalence fails to be true if Lipschitz continuity is relaxed, as it is shown in a couple of examples. Finally, we prove a theorem of approximation in W (1,1) and in the sense of mean curvature of C 2 graphs by polyhedral graphs. A number of examples illustrating different situations which can occur complete the work.

We study the pluripolar hull of the graph of a holomorphic function f, defined on a domain D ⊂ C outside a polar set A ⊂ D. This leads to a theorem that describes under what conditions f is nowhere extendible over A, while the graph of f over C A is not complete pluripolar.

The purpose of this paper is to show that the Markov inequality does not hold on the graph of holomorphic function.

The conjugate Gauss curvature functional is constructed. It is considered on the class of axisymmetrical surfaces generated by the curves represented by the graphs of functions whose domains are orthogonal to the axis of symmetry. The functional is applied to the variational study of equilibrium forms of liquid drops. It is responsible for the formation of intermediate layer between two phases, that of the liquid and of the gas. In the variational study presented the energies of surface tension, adhesion and of the gravitational forces are included. In contrast with classical approach it is not necessary to consider the adhesion’s angle as known beforehand. It can be calculated if the width of the intermediate layer is given.

In 1915, S. Bernstein proved that there is no nonflat minimal surface in R 3 which is described as the graph of a C 2-function on the total plane R 2 ([9]). This result was improved by many researchers in the field of differential geometry. E. Heinz studied nonflat minimal surfaces in R 3 which are the graph of functions on discs Δ R := {(x, y); x 2 + y 2 < R 2} and showed that there exists a constant C > 0 such that |K(0)| ≤ C/R 2 for the Gaussian curvature K(0) at the origin ([43]). After some related results were given by E. Hopf ([45]), J. C. C. Nitsche ([52]) and so on, in 1961 R. Osserman proved that the Gauss map of a nonflat complete regular minimal surface in R 3 cannot omit a set of positive logarithmic capacity in the Riemann sphere ([56]) and, in 1981 F. Xavier showed that the Gauss map of such a surface can omit at most six values. Moreover, in 1988 the author gave the best possible version of this, which asserts that the number of exceptional values of the Gauss map of a complete nonflat regular minimal surface is at most four. Recently, several related results were given by X. Mo and R. Osserman ([49]), S. J. Kao ([48]) and M. Ru ([60]).KeywordsRiemann SurfaceMinimal SurfaceMeromorphic FunctionCompact Riemann SurfaceHolomorphic CurveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A way to recover a closed convex hypersurface from its Gauss curvature is to find a positive function over Sn whose graph would represent the hypersurface in question. Then one is led to a nonlinear elliptic problem of Monge-Ampere type on Sn. Usually, geometric problems involving operators of this type are too complicated to be suggestive for a natural functional whose critical points are candidates for solutions of such problems. It turns out that for the problem indicated in the title, such a functional exists and has interesting geometric properties. With the use of this functional, we obtain new existence results for hypersurfaces with prescribed curvature as well as strengthen some that are already known. Introduction. In his book on convex polyhedrons Aleksandrov posed a general question of finding variational formulations and solutions to several geometric problems related to convex bodies [A, Chapter 7, §1, section 4]. As far as we know, until now such a solution for closed convex hypersurfaces is known only for the celebrated Minkowski problem. In this paper we present a variational solution to the following problem: Under what restrictions can a positive function K(X), X G Rn+1, n > 2, be realized as the Gauss-Kronecker curvature of some closed convex hypersurface in Pn+1? See Yau [Y, p. 683]. In [O], and also [D], it was shown that if K G Ck(Rn+1), k > 3, and some other conditions are satisfied then the hypersurface in question can be recovered as a graph of a smooth positive function p over a unit sphere Sn in Rn+1. The function p must satisfy on Sn a Monge-Ampere type equation of the form (*) (p2 4 |vp|2)-»/2-ip-2n+2det(-pHessp 4 2Vp x Vp 4 p2e) = ^ oeti e ) where Vp and Hess p denote correspondingly the gradient and Hessian of p in the standard metric e on Sn, and K is evaluated at the point X — (x,p(x)), x G Sn. Our main purpose in this paper is to construct and investigate a variational problem for which equation (*) is the Euler-Lagrange equation. The approach to solving (*) via variational calculus allows construction of solutions in the geometrically natural class of closed convex hypersurfaces not subject to any restrictions. The only smoothness requirement on the data is that the given function is continuous. The natural question when the solution to the variational problem is smooth will be treated in a separate publication. Received by the editors June 19, 1985. 1980 Mathematics Subject Classification. Primary 53C42; Secondary 49F22. 'Research supported by the National Science Foundation Grants MCS-8301904 and MCS8342997. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

We study a non-Hermitian chiral random matrix of which the eigenvalues are complex random variables. The empirical distributions and the radius of the eigenvalues are investigated. The limit of the empirical distributions is a new probability distribution defined on the complex plane. The graphs of the density functions are plotted; the surfaces formed by the density functions are understood through their convexity and their Gaussian curvatures. The limit of the radius is a Gumbel distribution. The main observation is that the joint density function of the eigenvalues of the chiral ensemble, after a transformation, becomes a rotation-invariant determinantal point process on the complex plane. Then, the eigenvalues are studied by the tools developed by Jiang and Qi [J. Theor. Probab. 30, 326 (2017); 32, 353 (2019)]. Most efforts are devoted to deriving the central limit theorems for distributions defined by the Bessel functions via the method of steepest descent and the estimates of the zero of a non-trivial equation as the saddle point.

Monodromies of generic real algebraic functions

Let A be a closed polar subset of a domain D in ℂ. We give a complete description of the pluripolar hull Γ D×ℂ * of the graph Γ of a holomorphic function defined on D∖A. To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.

Spacelike minimal surfaces which are graphs in [formula omitted

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire space of the Allen-Cahn equation Δu + u − u 3 = 0. In particular, we consider a minimal surface Γ in \({\mathbb {R}^9}\) which is the graph of a nonlinear entire function x 9 = F(x 1, . . . , x 8), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation in \({\mathbb {R}^9}\) which is monotone in the x9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and in eternal solutions to the flow by mean curvature for graphs.

Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns