Maria Karmanova

Area Formula for Graph Surfaces in sub-Lorentzian Geometry

The aim of the talk is to describe in sub-Lorentzian geometry surface images of graph mappings constructed using intrinsically Lipschitz mappings of Carnot groups. Sub-Lorentzian geometry can be considered as a sub-Riemannian version of well-known Minkowski geometry. The research in sub-Lorentzian geometry began only in recent years including its applications to physics. Moreover, structures with multi-dimensional time are also studied now.

We study graph mappings constructed from classes of intrinsically Lipschitz mappings of Carnot groups. Recall that intrinsically Lipschitz mappings on Carnot groups are Holder in the classical sense, nevertheless, they are differentiable in sub-Riemannian sense. Regarding the graphs of such mappings, it is known that they are not differentiable both in classical and in sub-Riemannian sense. We invent a special tool, the polynomial sub-Riemannian differentiability, that enables us to derive ``differential'' properties of graph mappings.

We introduce the following sub-Lorentzian distance on the image of graph-mappings.

Definition. If $w=\exp\Bigl(\sum\limits_{j=1}^{N+\widetilde{N}}y_jY_j\Bigr)(v)$ then squared sub-Lorentzian distance is equal to
$$
\mathfrak d^2_2(v,w)=\max\limits_{k=1,\ldots, \widehat{M}}\Biggl\{\operatorname{sgn}\Bigl(\sum\limits_{j:\,Y_j\in \widehat{V}_k^+}y_j^2-\sum\limits_{j:\,Y_j\in \widehat{V}_k^-}y_j^2\Bigr)\cdot\Bigl|\sum\limits_{j:\,Y_j\in \widehat{V}_k^+}y_j^2-\sum\limits_{j:\,Y_j\in \widehat{V}_k^-}y_j^2\Bigr|^{1/k}\Biggr\}.
$$
The corresponding sub-Lorentzian Hausdorff measure is defined by applying Caratheodory construction.

The main result is the theorem on sub-Lorentzian measure of graph surfaces.

Theorem. The following area formula
$$
\int\limits_{\Omega}{}^{SL}\mathcal J(\varphi, v)\,d\mathcal H^{\nu}(v)=\int\limits_{\varphi_{\Gamma}(\Omega)}\,d\ {}^{SL}\mathcal H^{\nu}_{\Gamma}(y)
$$
is valid with sub-Lorentzian Jacobian ${}^{SL}\mathcal J(\varphi, x)$ equal to
\begin{multline*}
\sqrt{\det\bigl(E_{\dim V_1}+\bigl(\widehat{D}\varphi^+\bigr)_{\widetilde{V}_1, V_1}^*(x)\bigl(\widehat{D}\varphi^+\bigr)_{\widetilde{V}_1, V_1}(x)-\bigl(\widehat{D}\varphi^-\bigr)_{\widetilde{V}_1, V_1}^*(x)\bigl(\widehat{D}\varphi^-\bigr)_{\widetilde{V}_1, V_1}(x)\bigr)}\times\\
\times
\prod\limits_{j=2}^M\sqrt{\det\bigl(E_{\dim V_j}-\bigl(\widehat{D}\varphi^-\bigr)_{\widetilde{V}_j, V_j}^*(x)\bigl(\widehat{D}\varphi^-\bigr)_{\widetilde{V}_j, V_j}(x)\bigr)}.
\end{multline*}

The publication was supported by the Ministry of Education and Science of the Russian Federation (the Project number 1.3087.2017/4.6).

XXXVIII Workshop on Geometric Methods in Physics	30.06-6.07.2019
VIII School on Geometry and Physics	24-28.06.2019

Event sponsored by:
University of Bialystok