Entire Minimal Graphs Research Articles

Bernstein and Half-Space Properties for Minimal Graphs Under Ricci Lower Bounds

Abstract In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold $M$ with Ricci curvature bounded from below. This enables us to show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on $M$ by exploiting a form of the Ahlfors–Khas’minskii duality in nonlinear potential theory.

On the asymptotic Plateau problem in [formula omitted

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space SL˜2(R) with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in SL˜2(R) contained between two bounded entire minimal graphs separated by vertical distance less than 1+4τ2π has multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in SL˜2(R).

A Bernstein-Type Theorem for Minimal Graphs of Higher Codimension via Singular Values

Instead on the slope function, we establish a condition on the singular values of the differential of a vector-valued function $f:\mathbb {R}^{n} \to \mathbb {R}^{m}$ which ensures that such f satisfying the minimal surface equations is affine linear. This is a Bernstein type theorem for entire minimal graphs of arbitrary dimension and codimension, improving the results in Jost and Xin (Calc. Var. PDE 9, 277–296, 1999) and Wang (Trans. Amer. Math. Soc. 355, 265–271, 2003). The proof depends on the superharmoncity of an auxiliary function and on conclusions from geometric measure theory.

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.

Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from $S$ onto $\Sigma$. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-d\sigma_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffeomorphism from $(S,g_1)$ to $(S,g_2)$, then $(S,g_1)$ is parabolic.

A sharp Bernstein-type theorem for entire minimal graphs

We consider entire solutions u to the minimal surface equation in $$\mathbb {R}^N$$ , with $$ N\ge 8,$$ and we prove the following sharp result: if $$N-7$$ partial derivatives $$ \frac{\partial u }{\partial {x_j}}$$ are bounded on one side (not necessarily the same), then u is necessarily an affine function.

Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(\kappa,\tau)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.

Strongly stable surfaces in sub-Riemannian 3-space forms

A surface of constant mean curvature (CMC) equal to H in a sub-Riemannian 3-manifold is strongly stable if it minimizes the functional area+2Hvolume up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian 3-manifolds. We also produce new examples of C1 complete CMC surfaces with empty singular set in the sub-Riemannian 3-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic 3-space M(−1). In relation to the Bernstein problem in M(−1) we discover strongly stable C∞ entire minimal graphs in M(−1) different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class C2 and non-empty singular set in the sub-Riemannian 3-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.

On the Bernstein Problem in the Three-dimensional Heisenberg Group

Abstract In this note we present a simple alternative proof for the Bernstein problem in the threedimensional Heisenberg group Nil3 by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch- Rosenberg diòerential.

Liouville Theorems for F-Harmonic Maps and Their Applications

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.

Bernstein type theorem for entire weighted minimal graphs in [formula omitted

Based on a calibration argument, we prove a Bernstein type theorem for entire minimal graphs over Gauss space Gn by a simple proof.

The half-space property and entire positive minimal graphs in $M \times \mathbb{R}$.

We show that a properly immersed minimal hypersurface in $M \times \mathbb{R}^+$ equals some $M \times\{c\}$ when $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with bounded curvature. If on the other hand, $M$ is not necessarily recurrent but has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over $M$.

A Free Boundary Problem Inspired by a Conjecture of De Giorgi

We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional , where V(u) is the characteristic function of the interval (−1, 1). This functional is a close relative of the scalar Ginzburg-Landau functional , where W(u) = (1 − u 2)2/2 is a standard double-well potential. According to a famous conjecture of De Giorgi, global critical points of J that are bounded and monotone in one direction have level sets that are hyperplanes, at least up to dimension 8. Recently, Del Pino, Kowalczyk and Wei gave an intricate fixed-point-argument construction of a counterexample in dimension 9, whose level sets “follow” the entire minimal non-planar graph, built by Bombieri, De Giorgi and Giusti (BdGG). In this paper we turn to the free boundary variant of the problem and we construct the analogous example; the advantage here is that of geometric transparency as the interphase {|u| <1} will be contained within a unit-width band around the BdGG graph. Furthermore, we avoid the technicalities of Del Pino, Kowalczyk and Wei's fixed-point argument by using barriers only.

Complete minimal graphs with prescribed asymptotic boundary on rotationally symmetric Hadamard surfaces

It is proved that if M is a rotationally symmetric Hadamard surface which is conformally equivalent to the hyperbolic disk then the asymptotic Dirichlet problem for the minimal surface equation is uniquely solvable for any continuous asymptotic boundary data. This result gives a partial answer of a question in Galvez and Rosenberg (Am J Math 132:1249–1273, 2010) about the existence of entire minimal graphs on Hadamard surfaces with sectional curvature possibly degenerating at infinity.

Half-space Theorems for Minimal Surfaces in Nil$_3$ and Sol$_3$

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol_3 that lies on one side of a special plane, then S is another special plane.

Construction of harmonic diffeomorphisms and minimal graphs

We study complete minimal graphs in ℍ x ℝ, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in ℍ. We give necessary and sufficient conditions on the lengths of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in ℍ x ℝ that are conformally the complex plane C. The vertical projection of such a graph yields a harmonic diffeomorphism from C onto ℍ, disproving a conjecture of Rick Schoen and S.-T. Yau.

The extrinsic curvature of entire minimal graphs in $\mathbb{H}^{2} \times \mathbb{R}$

We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in ℍ 2 × ℝ, ℍ 2 the hyperbolic plane.

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

We classify the entire minimal vertical graphs in the Heisenberg group Nil 3 endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil 3 , is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil 3 .

Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family of annuli is used to prove a vertical half-space theorem which is then applied to prove that each complete minimal graph in Nil3 is entire. Also, it is shown that the sister surface of an entire minimal graph in Nil3 is an entire constant mean curvature (CMC) 1 graph in H 2 ! R, and vice versa. This gives a classification of all entire CMC 1 graphs in H 2 ! R. Finally we construct properly embedded CMC 1 annuli in H 2 ! R.

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