2016/01/06 by Vicent Gimeno, Vicente Palmer
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Geometry and complex manifolds
paper · pdf · doi:10.1090/tran/6892
We explore the relation among volume, curvature and properness of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional isometric immersion in a Riemannian manifold. We show that, when the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Lp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm of the mean curvature vector is bounded for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m less-than-or-equal-to p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m ≤ p≤ ∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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> with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in <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> . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.