The Splitting Theorem in Non-Smooth Context

By Nicola Gigli. Memoirs of the American Mathematical Society, 2013

164 citations recorded by the source corpus.

<p> We prove that an infinitesimally Hilbertian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper D left-parenthesis 0 comma upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CD(0,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space containing a line splits as the product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an infinitesimally Hilbertian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper D left-parenthesis 0 comma upper N minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CD(0,N-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space. By ‘infinitesimally Hilbertian’ we mean that the Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript 1 comma 2 Baseline left-parenthesis upper X comma sans-serif d comma German m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">m</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W^{1,2}(X,\mathsf {d},\mathfrak {m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence. </p>

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Citations in the corpus

From volume cone to metric cone in the nonsmooth setting (2015, methodology): “Other relevant geometric properties, both on their own and for the purposes of the current paper, are the splitting theorem, proved in [12] by the first author, and the maximal diameter theorem, proved by Ketterer in [20].”
Characterization of Low Dimensional RCD*(K, N) Spaces (2015, unknown): “mply that Tan(X,d,m,x)consists of RCD∗(0,N)spaces for any point x ∈ suppm. One key tool that is reminiscent of smooth Riemannian setting is the splitting theorem: Theorem 2.8 (Splittingtheorem, Gigli [13,14]). Let (X,d,m)be an RCD∗(0,N)space with1 ≤ N <∞. Supposethat supp(m)containsaline. Then, (”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “, has been introduced in [4]. The Riemannian Curvature Dimension condition RCD(K,∞), has been introduced by L. Ambrosio, N. Gigli and G. Savar´e in [1]. Then the ﬁnite dimensional has been studied in [16, 17] and the precise RCD∗(K,N) has been deﬁned in [13] and [3] with two diﬀerent approaches. We ”
Local spectral convergence in RCD∗(K,N) spaces (2017, methodology): “; ǫ, (4.14) where eL i (z) := di(xLi,xi) − di(xLi,z). Moreover, we can take hi as the harmonic replacements of eL i in BR(xLi). Even though in the splitting theorem on RCD∗(0,N)-spaces established in [G13] the harmonic replacement were not needed (because a priori these spaces do not arise from an a”