A Canonical Infinitesimally Hilbertian Structure on Locally Minkowski Spaces

Abstract The aim of this paper is to show the existence of a canonical distance $$\mathsf d'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> defined on a locally Minkowski metric measure space $$(\textsf{X},\mathsf d,\mathfrak m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such that: $$\mathsf d'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> is equivalent to $$\mathsf d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> , $$(\textsf{X}, \mathsf d', \mathfrak m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is infinitesimally Hilbertian. This new regularity assumption on $$(\textsf{X}, \mathsf d,\mathfrak m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the reversible Finsler manifolds and it is actually strictly larger. The required distance $$\mathsf d'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> will be the intrinsic distance $$\mathsf d_\textsf{KS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>KS</mml:mi> </mml:msub> </mml:math> associated to the so-called Korevaar-Schoen energy , which is proven to be a quadratic form. In particular, we show that the Cheeger energy associated to the metric measure space $$(\textsf{X}, \mathsf d_\textsf{KS}, \mathfrak m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>KS</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is in fact the Korevaar-Schoen energy.