General rigidity principles for stable and minimal elastic curves

<jats:title>Abstract</jats:title> <jats:p>For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks’ and Sachkov’s rigidity principles for Euler’s elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed <jats:italic>p</jats:italic>-elasticae for all <jats:inline-formula id="j_crelle-2024-0018_ineq_9999"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <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-2024-0018_eq_0459.png" /> <jats:tex-math>{p\in(1,\infty)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and of stable pinned <jats:italic>p</jats:italic>-elasticae for <jats:inline-formula id="j_crelle-2024-0018_ineq_9998"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</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-2024-0018_eq_0457.png" /> <jats:tex-math>{p\in(1,2]}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our proof is based on a simple but robust “cut-and-paste” trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime <jats:inline-formula id="j_crelle-2024-0018_ineq_9997"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mn>3</m:mn> <m:mn>2</m:mn> </m:mfrac> <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-2024-0018_eq_0458.png" /> <jats:tex-math>{p\in(1,\frac{3}{2}]}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in which the second variation may not exist even for smooth variations.</jats:p>