Bernstein Type Theorem Research Articles

A Bernstein type theorem for minimal volume preserving maps

We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.

Some Aspects of the global Geometry of Entire Space-Like Submanifolds

We prove some Bernstein type theorems for entire space-like subma-nifolds in pseudo-Euclidean space and as a corollary, we give a new proof of the Calabi-Pogorelov theorem for Monge-Ampère equations.

Erratum to "Bernstein-type theorems in hypersurfaces with constant mean curvature" [An Acad Bras Cienc 72(2000): 301-310

2. A RESULT ON NODAL DOMAINS In this section we prove a result on the nodal domains of |φ| which will be needed in our proof of main theorems. We first need to recall the definition of nodal domains. Definition. An open domain D is called the nodal domain of a function f if f (x) = 0 for x ∈ intD and vanishes on the boundary of ∂D. We denote by N(f ) the number of disjoint bounded nodal domains of f . Now we have the following lemma which follows directly from Proposition 2.2 below. We want to thank the referee who provided the clearer proof of Proposition 2.2. Lemma 2.1. Let M be a hypersurface in Rn+1 with constant mean curvature H . Then ind(M) ≥ N(|φ|). (2.1)

Bernstein-type theorems in hypersurfaces with constant mean curvature

By using the nodal domains of some natural function arising in the study of hypersurfaces with constant mean curvature we obtain some Bernstein-type theorems.

Bernstein type theorems for higher codimension

We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improves previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds.

A Bernstein‐type theorem in hyperbolic spaces

The following Bernstein-type theorem in hyperbolic spaces is proved. Let ∑ be a non-zero constant mean curvature complete hypersurface in the hyperbolic space ℍn. Suppose that there exists a one-to-one orthogonal projection from ∑ into a horosphere. (1) If the projection is surjective, then ∑ is a horosphere. (2) If the projection is not surjective and its image is simply connected, then ∑ is a hypersphere.

Quasi-Sampling Sets for Analytic Functions in a Cone

We study analogues of sampling sets for analytic functions in cones ofCn. Cartwright-type and Bernstein-type theorems, previously known only for functions of exponential type inCn, are extended to the case of functions of arbitrary order in a cone.

Entropy, Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers en(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IRm and Sm. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers en(f) have the same asymptotics its ω(f, en(X)) for f e C(X). Finally, the results concerning functions f e M(X) and f e C(X) are transferred to operators with values in M(X) and C(X), respectively.

On Bernstein’s Theorem for Quasiminimal Surfaces Part II

Bernstein-type theorem is proved for surfaces with a quasiconformal Gauss map and with a growth condition for the total curvature.

Bernstein-type theorems for space-like surfaces with parallel mean curvature.

Article Bernstein-type theorems for space-like surfaces with parallel mean curvature. was published on January 1, 1997 in the journal Journal für die reine und angewandte Mathematik (volume 1997, issue 489).

On Bernstein type theorems concerning the growth of derivatives of entire functions

A subspace X of Ll,,(R) which is invariant under all left translation operators Tt, t E R, is called admissible if X is a Banach space satisfying the following properties: (i) If IIfnlIx 0, then there exists a subsequence (nk) such that fink (s) 0 almost everywhere. (ii) The group TX := {Ttlx : t E R} is a bounded strongly continuous group. In this case, let Cx := sup{HITtIx : t E RI. Typical admissible spaces are Co(R), BUC(R) and all spaces LP(R) for 1 0, e-6Itg(t) E L1 (R). With these definitions our main result goes as follows: Theorem 1. If g is an entire function of exponential type r such that its restriction to the real axis, denoted by gR, is subexponential and belongs to some admissible space X, then the derivative g' is also in X. Moreover, Ia + 9' IIX < (Oa2 + r2)1/2* CX 1lg1IIx for each real oz. This result yields as consequences and in a systematic way many new and old Bernstein type inequalities. 1. THE MAIN THEOREMS Recall that an entire function f is called to be of exponential type r if log M(r) lim Sup =7 r-woo r where M(r) := max{lf(z)I: IzI < r} is the maximum modulus of f. The classical Bernstein theorem [Bo, p.206, Theorem 11.1.2] states that if f is an entire function of exponential type r and is bounded on the real axis by M, then If'(t)I < TFM for all t c IR, or equivalently, IIf'IIIo < TIIflIIo for each entire function f of exponential type r. The latter is referred as to Bernstein's inequality. Many extensions are given, see [Bo, Chap. 11], [D-S], [Le, Chap. IV] and [K-S-T]. It is the purpose of Received by the editors August 16, 1995. 1991 Mathematics Subject Classification. Primary 30D20; Secondary 47D03, 47A10.

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.

Complex equilibrium measure and Bernstein type theorems for compact sets in ${\bf R}\sp n$

The aim of this paper is to refine and develop further some results from a paper of Bedford and Taylor [Trans. Amer. Math. Soc. 294 (1986), 705-717]. The main result, a Bernstein type theorem, is an improvement of the classical Bernstein inequality \[ |p’(x)| \leq (\deg p){(1 - {x^2})^{ - 1/2}}{(\left \| p \right \|_{[ - 1,1]}^2 - {p^2}(x))^{1/2}},\] from the interval $[ - 1,1]$ to the multivariate case.

Complex Equilibrium Measure and Bernstein Type Theorems for Compact Sets in R n

Complex Equilibrium Measure and Bernstein Type Theorems for Compact Sets in R n

Extreme convex set functions with many nonnegative differences

Where N is a finite set of the cardinality n and P the family of all its subsets, we study real functions on P having nonnegative differences of orders n - 2, n - 1 and n. Nonnegative differences of zeroth order, first-order, and second-order may be interpreted as nonnegativity, nonincreasingness and convexity, respectively. If all differences up to order n of a function are nonnegative, the set function is called completely monotone in analogy to the continuous case. We present a discrete Bernstein-type theorem for these functions with Möbius inversion in the place of Laplace one. Numbers of all extreme functions with nonnegative differences up to the orders n, n - 1 and n - 2, which is the most sophisticated case, and their Möbius transforms are found. As an example, we write out all extreme nonnegative nondecreasing and semimodular functions to the set N with four elements.

Bernstein Type Theorems for Compact Sets in Rn Revisited

In this paper we complete some results of (J. Approx. Theory69 (1992), 156-166) and give a geometrical approach to the multivariate Bernstein and Markov inequalities. The most interesting and slightly surprising result is a sharp Markov inequality for convex symmetric subsets of Rn formulated in geometrical language. A sharp inequality for gradients of polynomials extends an old Kellog result (Math. Z.27 (1927), 55-64), and it is also a partial positive answer to a question formulated by Wilhelmsen (J. Approx. Theory11 (1974), 216-220) in 1974.

Willmore submanifolds of the Möbius space and a Bernstein-type theorem

We study Willmore immersed submanifoldsf: Mm→Sn into then-Mobius space, withm≥2, as critical points of a conformally invariant functionalW. We compute the Euler-Lagrange equation and relate this functional with another one applied to the conformal Gauss map of immersions intoSn. We solve a Bernestein-type problem for compact Willmore hypersurfaces ofSn, namely, if ∃a ∈ℝn+2 such that ≠ 0 onM, whereγf is the hyperbolic conformal Gauss map and is the Lorentz inner product ofℝn+2, and iff satisfies an additional condition, thenf(M) is an (n−1)-sphere.

On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space

Let RV1 be the (n+l)-dimensional Minkowski space, that is, Rn+l with the Lorentz metric < ,)―(dx1)2+ ・・・+(dxn)2―(dxn+1)z. It has been known that in R +1 hyperplanes are the only complete space-like hypersurfaces whose mean curvatures are zero. This Bernstein type theorem was proposed by Calabi, and solved by him [3] (for n^4) and by Cheng and Yau [5] (for all n) (see alsoIshihara [10] or Nishikawa [14]). On the other hand, for complete space-like hypersurfaces of nonzero constant mean curvature in Rnx+ there are many nonlinear examples constructed by Treibergs [18], Hano and Nomizu [7], Isbihara and Hara [11] and others. In his recent paper, Palmer [17] discussed the Gauss map of a complete space-like hypersurface of constant mean curvature in 1??+1 and showed a condition for the hypersurface to be a hyperplane. This is a result analogous to the one obtained by Hoffman, Osserman and Schoen [9], who proved that the normals to a complete surface of constant mean curvature in the 3-dimensional Euclidean space Es cannot lie in a closed hemisphere of S*, unless the surface is a plane or a right circular cylinder. Note that a right circular cylinder is the simplest example of a complete non-umbilical surface of constant mean curvature in Es. In R+1 the simplest example of a complete non-umbilical space-like hypersurface of constant mean curvature is given by the following:

Optimal approximation and growth of solutions to a class of elliptic partial differential equations

Function theoretic methods derive Bernstein type theorems for a class of second order elliptic partial differential equations to determine the disk of regularity and growth of solutions at the boundary.

Berstein Type Theroms Without Graphic Condition

Abstract. We prove Bernstein type theorems for minimal n-submanifolds in R n+p with flatnormal bundle under certain growth conditions for p ≥ 2 and n ≤ 5, as well as for arbitrary n andp = 1. When M is a graphic minimal hypersurface we recover the known result.Key words. minimal submanifold, flat normal bundleAMS subject classifications. 49Q05, 53A07, 53A10 1. Introduction. There are various generalizations of the Bernstein theorem .For an entire minimal graph M = (x,f(x)) of dimension n ≤ 7 in R n+1 the problemhas been settled in [Si], [B-G-G]. If there is no dimension limitation, we have resultsin [M], [C-N-S], [Ni] and [E-H] under the growth condition on the function f.For general minimal hypersurfaces, a natural condition is stable, or even oneswith finite index. In this situation [C-S-Z] and [L-W] conclude that they have onlyone end or finitely many ends, respectively.Higher codimensional Bernstein problem becomes more complicated. In [H-J-W] Moser’s result has been generalized to minimal graphs of higher codimension.Recently, we obtained better results under a bound for the slope of the vector-valuedfunctions f which is independent of the dimension and codimension [J-X1] [J-X2](alsosee [Wm]. On the other hand, the counter example of [L-O] prevents us from goingfurther.For general minimal surfaces in a Euclidean space, so-called parametric case, theBernstein theorem was generalized to the beautiful theory of the value distribution ofits Gauss image [Os] [Xa] [F].In the present article we study minimal n-submanifolds of R