A Notable Family of Entire Intrinsic Minimal Graphs in the Heisenberg Group which are not Perimeter Minimizing

Abstract

Abstract. One of the main objectives of this paper is to unravel a new interesting phenomenon of the sub-Riemannian Bernstein problem with respect to its Euclidean ancestor, with the purpose of also indicating a possible line of attack toward its solution. We show that the global intrinsic graphs (1.2) are unstable critical points of the horizontal perimeter. As a consequence of this fact, the study of the stability acquires a central position in the problem itself.

Generalized Mean Curvature Flow in Carnot Groups

“…In view of this new difficulty, in the present paper while we are able to prove existence for the general flow described in (1.3), at the moment we can prove comparison principles and uniqueness only for a special 1 Closely linked to the study of mean curvature flow, the analysis of minimal surfaces in the sub-Riemannian setting has recently seen great activity [32], [56], [15], [16], [33], [22], [5] and [54]). …”

The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations

“…Example 2.2). These considerations (see also [18]) suggest that the right counterpart of the classical Bernstein problem in the Heisenberg setting is Again, as in the Euclidean setting, the answer seems to depend on the dimension n of the space; however, new and unexpected phenomena seem to arise, e.g. the fact that we have solutions to (1.16) in H 1 which are not area minimizing.…”

“…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 make precise below, see [11,12,18,28,47].…”

“…In this paper we want to rephrase the Bernstein problem in H n assuming again the notion of vertical plane but replacing the notion of Euclidean graph with another one more intrinsic with respect to the CC geometry, which turned to be very useful in GMT (see also [18]). …”

An intrinsic measure for submanifolds in stratified groups

“…Here we address the reader to some relevant papers [1], [2], [5], [4], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [21], [25], [26], [28], [29], [22], [23], [30], [31], [33], [34], [36], [37], [39], [43], [44], [45], [46] and the reference therein.…”