On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group

Ferrari, Fausto; Liu, Qing; Manfredi, Juan J.

doi:10.1142/s0219199713500272

Cited by 22 publications

(42 citation statements)

References 27 publications

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“…Mean curvature flow in the setting of Carnot group has been studied by [6] and [12]. See also the recent [20] as well as [21] for a probabilistic interpretation of the flow.…”

Section: 2mentioning

confidence: 99%

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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Abstract. We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ ǫ collapsing to a subRiemannian metric σ 0 as ǫ → 0. We establish C k,α estimates for this flow, that are uniform as ǫ → 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in [10] to the total variation flow. IntroductionThe mean curvature flow is the motion of a surface where each points is moving in the direction of the normal with speed equal to the mean curvature In the case where the evolution of graphs S t = {(x, u(x, t))} ⊂ R n × R is considered, then, provided enough regularity is assumed, the function u satisfies the equationGiven appropriate boundary/initial conditions, global in time solutions asymptotically converge to minimal graphs.In this paper we study long time existence of graph solutions of the mean curvature flow in a special class of degenerate Riemannian ambient spaces: The so-called sub-Riemannian Hörmander type setting [22], [40]. In particular we will focus on a class of Lie groups endowed with a metric structure (G, σ 0 ) that arises as limit of collapsing left-invariant tame Riemannian structures (G, σ ǫ ).Our approach to the existence of global (in time) smooth solutions is based on a Riemannian approximation scheme. We study graph solutions of the mean curvature flow in the Riemannian spaces (G, σ ǫ ) where G is a group and σ ǫ is a family of Riemannian metrics that 'collapse' as ǫ → 0 to a sub-Riemannian metric σ 0 in G.

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“…Mean curvature flow in the setting of Carnot group has been studied by [6] and [12]. See also the recent [20] as well as [21] for a probabilistic interpretation of the flow.…”

Section: 2mentioning

confidence: 99%

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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“…On the other hand, existence of solutions in H has been recently treated in different works [1,10,24], etc, employing different procedures including approximation schemes, Perron's method, stochastic games, among others. In the present article, we study the problem of existence of viscosity solutions to (1.1) via the celebrated Perron's method adapted to the theory of viscosity solutions (see [14]) and to the Heisenberg group.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and uniqueness results for linear second-order equations in the Heisenberg group

Ochoa¹,

Ruiz²

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this manuscript, we prove uniqueness and existence results of viscosity solutions for a class of linear second-order equations in the Heisenberg group. We state uniqueness by proving a comparison result to our class of equations, and existence via an application of Perron's method adapted to our framework. We also provide the explicit construction of the appropriate sub-and supersolutions employed by Perron's method for a variety of domains in the Heisenberg group.

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“…Very little is known about both flows in the subRiemannian setting and as far as we know the results in [14,17] are the first to establish existence of long time smooth flows. For other contributions to this topics, from different points of view, we recall the recent work in [35,36]. …”

Section: Here We Have Denoted By B the Balls Related To The D Distancmentioning

confidence: 99%

Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

Capogna

Citti

2015

Bull. Math. Sci.

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We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea (Math Ann 357(3):1175-1198, 2013) and Manfredini (Anal Geom Metric Spaces 1:255-275, 2013) concerning stability of doubling properties, Poincare' inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of Hörmander vector fields.

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On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group

Cited by 22 publications

References 27 publications

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

Existence and uniqueness results for linear second-order equations in the Heisenberg group

Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

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