Nonabelian Yang–Mills–Higgs and Plateau’s problem in codimension three

<jats:p> We investigate the asymptotic behavior of the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SU</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2026-0038_ineq_0001.png"/> <jats:tex-math>\operatorname{SU}(2)</jats:tex-math> </jats:alternatives> </jats:inline-formula> -Yang–Mills–Higgs energy <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>E</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="0.111em">=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mo rspace="0em">∫</m:mo> <m:mi>M</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>A</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">Φ</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>A</m:mi> </m:msub> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2026-0038_ineq_0002.png"/> <jats:tex-math>E(\Phi,A)=\int_{M}\lvert d_{A}\Phi\rvert^{2}+\lvert F_{A}\rvert^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi’s Γ-convergence. More precisely, for a compact manifold with boundary 𝑀 and any family of pairs <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">Φ</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="normal">Ω</m:mi> <m:mn>0</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>;</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2026-0038_ineq_0003.png"/> <jats:tex-math>\Phi_{m}\in\Omega^{0}(M;\mathfrak{su}(2))</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="normal">Ω</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>;</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2026-0038_ineq_0004.png"/> <jats:tex-math>A_{m}\in\Omega^{1}(M;\mathfrak{su}(2))</jats:tex-math> </jats:alternatives> </jats:inline-formula> indexed by a <jats:italic>mass</jats:italic> parameter <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2026-0038_ineq_0005.png"/> <jats:tex-math>m\to\infty</jats:tex-math> </jats:alternatives> </jats:inline-formula> , satisfying </jats:p> <jats:p> <jats:disp-formula id="j_crelle-2026-0038_eq_9999"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mi>E</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="normal">Φ</m:mi> <m:mi>m</m:mi> </m:msub> <m: