arXiv subjects
Publications and source records attributed to András Domokos.
In this paper we point out an interesting geometric structure of nonnegative metric curvature emerging from the hyperspaces of decomposable, non-locally connected homogeneous continua, where "smooth" and "non-smooth" partitions live together, similarly to the macroscopic and the quantum realms of the Universe.
In this paper we prove uniform convergence of approximations to $p$-harmonic functions by using natural $p$-mean operators on bounded domains of the Heisenberg group $\mathbb{H}$ which satisfy an intrinsic exterior corkscrew condition. These domains include Euclidean $C^{1,1}$ domains.
In this paper we introduce a new type of exponential map in semi-simple compact Lie groups, which is related to the sub-Riemannian geometry generated by the orthogonal complement of a Cartan subalgebra in a similar way to how the group exponential map is related to the Riemannian geometry.
Let the vector fields $X_1, ... , X_{6}$ form an orthonormal basis of ${\mathcal H}$, the orthogonal complement of a Cartan subalgebra (of dimension $2$) in SU(3). We prove that weak solutions $u$ to the degenerate subelliptic $p$-Laplacian $$ Δ_{\mathcal{H},{p}} u(x)=\sum_{i=1}^{6} X_i^{*}\left(|\nabla_{\hspace{-0.1cm} {\mathcal H}} u|^{p-2}X_{i}u \right) =0,$$ have Hölder continuous horizontal derivatives $\nabla_{\hspace{-0.1cm}{\mathcal H}} u=(X_1u, \ldots, X_{6}u)$ for $p\ge 2$. We also prove that a similar result holds for all compact connected semisimple Lie groups.
Length spectra for Riemannian metrics are well studied, while sub-Riemannian length spectra have been largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by restricting its Killing form to the sum of the root spaces. Surprisingly, the shortest loops are the same in both the Riemannian and sub-Riemannian cases. We provide specific calculations for SU(2) and SU(3).