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

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 \sigma_\e collapsing to a subRiemannian metric $\sigma_0$ as \e\to 0. We establish $C^{k,\alpha}$ estimates for this flow, that are uniform as \e\to 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 \cite{CCM3} to the total variation flow.Comment: arXiv admin note: text overlap with arXiv:1212.666

Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups

In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics \sigma_\e which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as \e\to 0. The main new contribution are Gaussian-type bounds on the heat kernel for the \sigma_\e metrics which are stable as \e\to 0 and extend the previous time-independent estimates in \cite{CiMa-F}. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in (G,\s_\e). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as \e\to 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (\e=0), which in turn yield sub-Riemannian minimal surfaces as $t\to \infty$.Comment: We have corrected a few typos and added a few more details to the proof of the Gaussian estimate

Response to Letter by Dr. Santana-Penin

n/

Regularity of Mean Curvature Flow of Graphs On Lie Groups Free Up To Step 2

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 ε collapsing to a subRiemannian metric σ0 as ε → 0. We establish Ckα estimates for this flow, that are uniform as ε → 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 Capogna et al. (2013) to the total variation flow

Regularity of Non-characteristic Minimal Graphs In the Heisenberg Group ℍ\u3csup\u3e1\u3c/sup\u3e

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are apriori estimates on the solutions of the approximating Riemannian PDE and the ensuing C∞ regularity of the sub-Riemannian minimal surface along its Legendrian foliation

Smoothness of Lipschitz Minimal Intrinsic Graphs In Heisenberg Groups ℍ\u3csup\u3en\u3c/sup\u3e, N \u3e 1

We prove that Lipschitz intrinsic graphs in the Heisenberg groups ℍn, with n \u3e 1, which are vanishing viscosity solutions of the minimal surface equation, are smooth and satisfy the PDE in a strong sense

Regularity of non-characteristic minimal graphs in the Heisenberg group H1H^1

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^{\infty}$ regularity of the sub-Riemannian minimal surface along its Legendrian foliation

WISE: A Semantic and Interoperable Web of Things Architecture for Smart Environments

The rapid proliferation of Internet of Things devices has led to a number of different standards and technologies which offer novel and exciting services. One of the key aspect of the Internet of Things is its ubiquitness, as devices may spontaneously form networks and leave them possibly in short time frames. This is the case of Smart Environments such as Smart Homes, in which users carry a set of devices like wearables and mobile applications to monitor their behavior and provide contextual services. However, the interoperability and seamless interaction of different devices is yet to be fully realized. In this paper we propose WISE, a framework that leverages the Web of Thing architecture and Semantic technologies to overcome technical and conceptual interoperability difficulties and enables the creation of cooperative Smart Environments that self-adapt on the basis of users' preferences. The use of Semantic technologies enables to understand which devices can provide the needed affordances to meet the user preferences, while the WoT architecture is leveraged to access devices in a standardized manner. We also propose a reference implementation based on off-the-shelf devices which demonstrate the feasibility of WISE