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<p>Let <i>P</i> be a convex body containing the origin in its interior. We study a real Monge–Ampère equation with singularities along <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\partial P\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation> which is Legendre dual to a certain free boundary Monge–Ampère equation. This is motivated by the existence problem for complete Calabi–Yau metrics on log Calabi–Yau pairs (<i>X</i>,&#xa0;<i>D</i>) with <i>D</i> an ample, simple normal crossings divisor. We prove the existence of solutions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{\infty }(P)\cap C^{1,\alpha }(\overline{P})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mover> <mi>P</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and establish the strict convexity of the free boundary. When <i>P</i> is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension 1 faces of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\partial P\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A Free Boundary Monge–Ampère Equation and Applications to Complete Calabi–Yau Metrics

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Tristan C. Collins,
  • Freid Tong,
  • Shing-Tung Yau

摘要

Let P be a convex body containing the origin in its interior. We study a real Monge–Ampère equation with singularities along \(\partial P\) ∂ P which is Legendre dual to a certain free boundary Monge–Ampère equation. This is motivated by the existence problem for complete Calabi–Yau metrics on log Calabi–Yau pairs (X, D) with D an ample, simple normal crossings divisor. We prove the existence of solutions in \(C^{\infty }(P)\cap C^{1,\alpha }(\overline{P})\) C ∞ ( P ) ∩ C 1 , α ( P ¯ ) , and establish the strict convexity of the free boundary. When P is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension 1 faces of \(\partial P\) ∂ P .