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<p>Let (<i>M</i>, <i>g</i>) be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian <i>L</i>(<i>x</i>, <i>v</i>): <i>TM</i> → ℝ defined by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L(x,\, v) := {1 \over 2}gx(v, \, v)-\omega(v) + c\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>g</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>v</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </math></EquationSource> </InlineEquation>, where <i>c</i> ∈ ℝ and <i>ω</i> is a closed 1-form. From the perspective of differential geometry, we estimate the Laplacian of the weak Kolmogorov-Arnold-Moser (KAM) solution <i>u</i> to the associated Hamilton-Jacobi equation <i>H</i>(<i>x</i>, <i>du</i>) = <i>c</i>[<i>L</i>] in the barrier sense. This analysis enables us to prove that each weak KAM solution <i>u</i> is a constant if and only if <i>ω</i> is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and the Mañé Lagrangian.</p>

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A geometric approach to the Mather quotient problem

  • 期刊论文

  • 发表日期：

    2025年12月3日

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  • Wei Cheng,
  • Wenxue Wei

摘要

Let (M, g) be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian L(x, v): TM → ℝ defined by \(L(x,\, v) := {1 \over 2}gx(v, \, v)-\omega(v) + c\) L ( x , v ) := 1 2 g x ( v , v ) − ω ( v ) + c , where c ∈ ℝ and ω is a closed 1-form. From the perspective of differential geometry, we estimate the Laplacian of the weak Kolmogorov-Arnold-Moser (KAM) solution u to the associated Hamilton-Jacobi equation H(x, du) = c[L] in the barrier sense. This analysis enables us to prove that each weak KAM solution u is a constant if and only if ω is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and the Mañé Lagrangian.