Minimal surfaces in pseudohermitian geometry

By Jih-Hsin Cheng, Jenn-Fang Hwang, A. Malchiodi, Paul Yang. 2004

175 citations recorded by the source corpus.

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the xy-plane) in the Heisenberg group H1. In H1, identified with the Euclidean space R3, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S3. This fact continues to hold when S3 is replaced by a general pseudohermitian 3-manifold. Mathematics Subject Classification (2000): 35L80 (primary); 35J70, 32V20, 53A10, 49Q10 (secondary).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3b5263970bda65f91def664c4520af6a1100388a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space (2007, background): “studied in the last few years (see [BSV, CHMY , GaPa, RiRo] and references therein).”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, background): “Several first variation formula have been independently established in receent years, see for instance [18], [7], [8], [4], [32], [33], [27], [28], [35], [36].”
The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, methodology): “The Bernstein problem in the framework of CC spaces (see Definition (2.2) below) has been recently studied for the first Heisenberg group H 1 with suitable assumptions we will precise below (see [27],[ 12 ], [11],[45] and [17]).”
Properly embedded and immersed minimal surfaces in the Heisenberg group (2004, methodology): “o have no singular points. Theorem C.(1.2) has no singular points if and only if θ′(t) ≥ 0 for all t∈ R. Note that (1.1) has no isolated singular points, but does have one (connected) singular curve ([CHMY]). By taking θ(t) = −cot−1(t) and x0 = y0 = 0 in (1.2), we obtain y= xz. This is an entire p-m”
Existence and uniqueness for P-area minimizers in the Heisenberg group (2006, methodology): “In [3], we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry.”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “To quote a few: H0 can be defined in terms of the first variation of the area functional [20, 29, 9, 36, 40, 31, 7], as horizontal divergence of the horizontal unit normal or as limit of the mean curvatures Hε in the Riemannian metrics gε as ε → 0.”
H-minimal graphs of low regularity in H1 (2006, methodology): “The first variation formula has been explored in a variety of settings by a number of authors including the aforementioned paper of Cheng, Hwang, Malchiodi and Yang ([5]), Danielli, Garofalo and Nhieu ([7]), the author ([16]), Bonk and Capogna ([1]), Ritoré and Rosales ([17]).”
Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity (2008, methodology): “t;1,has been recently obtained by Capogna, Citti and Manfredini [6]. Characterization in H1 of solutions of the Bernstein problem for C2 surfaces has been obtained by Cheng, Hwang, Malchiodi and Yang [8], and Ritor´e and Rosales [29] for tgraphs, and by Barone Adessi, Serra Cassano and Vittone [4] a”
Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, background): “This vector is called by some authors characteristic direction and plays an important role in the study of minimal surfaces in H and in the study of the properties of the characteristic set char( ), see for instance [12,13] and [11].”
TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS (2007, background): “Calculus and Analysis in stratified groups have been developed by several authors with a vast literature, here we mention a list of references for a rough overview that is certainly far from being complete, [1], [3], [4], [7], [8], [9], [11], [ 12 ], [13], [14], [15], [20], [21], [22], [24], [28], [”
Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group (2007, background): “Starting from the work [6] (see also [5]), we studied the subject from the viewpoint of partial differential equations and that of differential geometry.”
Convex isoperimetric sets in the Heisenberg group (2006, background): “6) has been studied by several authors under different regularity assumptions (besides the previous references, see also [4–6, 19, 20]).”