<p>We extend Struwe’s result (Acta Math. 160(1–2):19–64, 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(0, \pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Specifically, let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> be a surface in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> diffeomorphic to the sphere, and let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Sigma '$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Σ</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> be a convex surface enclosing <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>. Given <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\tau \in (-1, 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and a constant <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$H \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> below the infimum of the mean curvature of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Sigma '$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Σ</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>, we show that, for almost every <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$r \in (0, 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, in the region enclosed by <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Sigma '$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Σ</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>, there exists a branched immersed disk with constant mean curvature <i>rH</i> whose boundary meets <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> at an angle with cosine value <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$r\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mi>τ</mi> </mrow> </math></EquationSource> </InlineEquation>. Moreover, the constant mean curvature disks we construct have index at most 1.</p>

Existence of Constant Mean Curvature Disks in $\mathbb {R}^3$ with Capillary Boundary Condition

期刊论文
发表日期：

2026年3月27日

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Da Rong Cheng

摘要

We extend Struwe’s result (Acta Math. 160(1–2):19–64, 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in $(0, \pi )$ $(0, π)$ . Specifically, let $\Sigma $ $Σ$ be a surface in $\mathbb {R}^3$ $R^{3}$ diffeomorphic to the sphere, and let $\Sigma '$ $Σ^{'}$ be a convex surface enclosing $\Sigma $ $Σ$ . Given $\tau \in (-1, 1)$ $τ \in (- 1, 1)$ and a constant $H \ge 0$ $H \geq 0$ below the infimum of the mean curvature of $\Sigma '$ $Σ^{'}$ , we show that, for almost every $r \in (0, 1)$ $r \in (0, 1)$ , in the region enclosed by $\Sigma '$ $Σ^{'}$ , there exists a branched immersed disk with constant mean curvature rH whose boundary meets $\Sigma $ $Σ$ at an angle with cosine value $r\tau $ $r τ$ . Moreover, the constant mean curvature disks we construct have index at most 1.

Existence of Constant Mean Curvature Disks in \(\mathbb {R}^3\) with Capillary Boundary Condition

摘要