Monotonicity Formulas for Harmonic Functions in $$\textrm{RCD}(0,N)$$ Spaces
Nicola Gigli, Ivan Yuri Violo
Abstract
Abstract We generalize to the $$\textrm{RCD}(0,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>RCD</mml:mtext><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in $$\textrm{RCD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>RCD</mml:mtext></mml:math> spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in $$\textrm{RCD}(K,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>RCD</mml:mtext><mml:mo>(</mml:mo><mml:mi>K</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.