A Sobolev gradient flow for the area-normalised Dirichlet energy of H ¹ maps

In this article we study the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {H¹} -gradient flow for the energy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {E[X]} given by the quotient of the Dirichlet energy and the signed enclosed area of an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {H¹} map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>𝕊</m:mi> <m:mo>→</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> {X:S²} . We prove that solutions with initially positive signed enclosed area exist eternally, and converge as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {t→∞} to a (possibly multiply-covered) circle. In this way we recover an improved parametrised isoperimetric inequality for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {H¹} maps.