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The Lie groupoid analogue of a symplectic Lie group

David N. Pham (2021)

Archivum Mathematicum

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a t -symplectic Lie groupoid; the “ t " is motivated by the fact that each target fiber of a t -symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid 𝒢 ⇉ M , we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid...

The mean curvature measure

Quiyi Dai, Neil S. Trudinger, Xu-Jia Wang (2012)

Journal of the European Mathematical Society

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...

The mean curvature of a Lipschitz continuous manifold

Elisabetta Barozzi, Eduardo Gonzalez, Umberto Massari (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The paper is devoted to the description of some connections between the mean curvature in a distributional sense and the mean curvature in a variational sense for several classes of non-smooth sets. We prove the existence of the mean curvature measure of ∂ ⁡ E by using a technique introduced in [4] and based on the concept of variational mean curvature. More precisely we prove that, under suitable assumptions, the mean curvature measure of ∂ ⁡ E is the weak limit (in the sense of distributions) of the mean...

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