Weighted monotonicity theorems and applications to minimal surfaces of ℍⁿ and 𝕊ⁿ

2023/03/22 by Manh Tien Nguyen
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Black Holes and Theoretical Physics #Nonlinear Partial Differential Equations

paper · doi:10.1090/tran/8949

Abstract

We prove that in a Riemannian manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Sⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the hyperbolic space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Hⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as the distance function, the Euclidean coordinates of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Rⁿ⁺¹</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Minkowskian coordinates of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n comma 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R^n,1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper O left-parenthesis n comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</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">SO(n,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-distinct unweighted monotonicity theorems. From these, we obtain upper bounds of the Graham–Witten renormalised area of a minimal surface in term of its ideal perimeter measured under different metrics of the conformal infinity. Other applications include a vanishing result for knot invariants coming from counting minimal surfaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Hⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a quantification of how antipodal a minimal submanifold of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript n"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Sⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has to be in term of its volume.

Citations

• Heat Equations on Minimal Submanifolds and Their Applications
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products

Related

• Weighted monotonicity theorems and applications to minimal surfaces in ℍⁿ and Sⁿ
• Bismut Ricci flat generalized metrics on compact homogeneous spaces
• Mean curvature, volume and properness of isometric immersions
• Approximating topological metrics by Riemannian metrics
• A Schwarz lemma for meromorphic functions and estimates for the hyperbolic metric