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Международная молодежная конференция «Геометрия и управление»
18 апреля 2014 г. 12:00, г. Москва, МИАН

Geometric and Analytic Properties of Carnot–Carathéodory Spaces under Minimal Smoothness

Sergey Basalaev

Novosibirsk State University, Novosibirsk, Russia
Flash Video 158.2 Mb
Flash Video 946.9 Mb
MP4 158.2 Mb
Adobe PDF 209.0 Kb
Adobe PDF 95.6 Kb

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Sergey Basalaev

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Аннотация: We describe geometric and analytical results in the theory of non-holonomic spaces under minimal smoothness, which we define following works [1, 2].
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Definition. Fix a connected Riemannian $C^\infty$-manifold $\mathbb{M}$ of topological dimension $N$. The manifold $\mathbb{M}$ is called the Carnot–Carathéodory space if the tangent bundle $T\mathbb{M}$ has a filtration
$$ H \mathbb{M} = H_1 \mathbb{M} \subsetneq H_2 \mathbb{M} \subsetneq …\subsetneq H_M \mathbb{M} = T \mathbb{M} $$
by subbundles such that every point $x_0 \in \mathbb{M}$ has a neighborhood $U(x_0) \subset \mathbb{M}$ equipped with a collection of $C^1$-smooth vector fields $X_1, …, X_N$ enjoying the following two properties:
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(1) At every point $x \in U(x_0)$ we have a subspace
$$ H_i \mathbb{M} (x) = H_i (x) = \mathrm{span} \{ X_1(x), …, X_{\dim H_i}(x) \} \subset T_x \mathbb{M} $$
of the dimension $\dim H_i$ independent of $x$, $i = 1, …, M$.
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(2) The inclusion $[H_i, H_j] \subset H_{i+j}$ holds for $i+j \leq M$.
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Moreover, the Carnot–Carathéodory space is called the Carnot manifold if the following third condition holds:
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(3) $H_{j+1} = \mathrm{span} \{ H_j, [H_1, H_j], …, [H_k, H_{j+1-k}] \}$, where $k = \lfloor \tfrac{j+1}{2} \rfloor$ for $j = 1, …, M-1$.
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Since it is not a priori known whether Carnot manifolds carry Carnot–Carathéodory metric, the Nagel–Stein–Wainger “Box” metric $d_\infty(x,y)$ is used instead in their study. Using results on fine properties of Carnot–Carathéodory spaces [2] we show that Carnot–Carathéodory metric is well-defined proving an analogue of Carathéodory– Rashevskiĭ–Chow theorem.
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Theorem [3]. 1) For every point $g \in \mathbb{M}$ there is a neighborhood $U$ and $C > 0$ such that every point $x \in U$ can be represented as
$$ x = \exp(a_L X_{i_L}) \circ …\circ \exp(a_1 X_{i_1})(g) $$
with $i_k \in \{ 1, …, \dim H_1 \}$ and $|a_k| \leq C d_\infty(x,g)$ for $k = 1, …, L$. Here $L = L(\mathbb{M})$ does not depend on the points $g$, $x$.
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2) In a connected Carnot manifold any two points can be joined by an absolutely continuous curve consisting of finitely many segments of integral lines of vector fields $X_1, …, X_{\dim H_1}$.
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This result in turn was utilized in [4] to prove local equivalence of “Box” metric $d_\infty$ and Carnot–Carathéodory metric $d_{cc}$ which immediately implies that:
$\bullet$ locally there is $C>0$ such that $B(x, C^{-1} r) \subset \mathrm{Box}(x,r) \subset B(x, Cr)$;
$\bullet$ the Hausdorff dimension of $\mathbb{M}$ is $\nu = \sum\limits_{k=1}^N \deg X_k$;
$\bullet$ the Hausdorff measure $\mathcal{H}^\nu$ is locally doubling.
As an application of these results to a theory of Sobolev spaces we obtain the Poincaré inequality for John domains in Carnot manifolds.
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Theorem [5]. Let $x_0 \in \mathbb{M}$ and $1 \leq p < \infty$. There are $C_p > 0$ and $r_0 > 0$ such that for every John domain $\Omega \subset B(x_0, r_0)$ of class $J(a,b)$ and every $f \in C^\infty(\overline{\Omega})$ we have
$$ \Vert f - f_\Omega \Vert_{L_p(\Omega)} \leq (\tfrac{b}{a})^\nu \mathrm{diam} (\Omega) \Vert (X_1 f, …, X_{\dim H_1} f) \Vert_{L_p(\Omega)} $$
where $f_\Omega = \tfrac{1}{|\Omega|} \int_\Omega f$ and $\nu$ is the Hausdorff dimension of $\mathbb{M}$.

Материалы: slides.pdf (209.0 Kb), abstract.pdf (95.6 Kb)

Язык доклада: английский

Список литературы
  1. Vodopyanov S. K, Geometry of Carnot–Carathéodory spaces and differentiability of mappings. In: The Interaction of Analysis and Geometry, Amer. Math. Soc. Providence, 2007. P. 247–302.
  2. Karmanova M., Vodopyanov S, Geometry of Carnot–Carathéodory spaces, differentiability, coarea and area formulas. In: Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser, Basel, 2009. P. 233–335.
  3. Basalaev S. G., Vodopyanov S. K, Approximate differentiability of mappings of Carnot–Carathéodory spaces. // Eurasian Math. J. 2013. V. 4, N. 2. P. 10–48.
  4. Karmanova M., Vodopyanov S, On local approximation theorem on equiregular Carnot–Carathéodory spaces. In: Proc. INDAM Meeting on Geometric Control and Sub-Riemannian Geometry. Springer INDAM Ser., 2014. V. 5. P. 241–262.
  5. Basalaev S, The Poincaré inequality for $C^{1,\alpha}$-smooth vector fields. // Siberian Math. J. 2014. V. 55, N. 2. P. 210–224.

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