Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript alpha"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">u_\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a sequence of smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-harmonic maps from a compact Riemann surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with boundary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper M"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a compact Riemannian manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with free boundary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript alpha Baseline left-parenthesis partial-differential upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u_\alpha (\partial M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a supporting submanifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with uniformly bounded <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-energy. If the target manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript upper K minus 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{K-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that there is no energy loss for such a sequence of maps during the blow-up process as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha down right-arrow 1"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">↘</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \searrow 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, the image of the weak limit map and bubbles is a connect set. Also, the case of Dirichlet boundary is considered.

The qualitative behavior for 𝛼-harmonic maps from a surface with boundary into a sphere

. Dirac-harmonic maps ( f,ϕ ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ ( Σ M ⊗ f ∗ TN ) and they are deﬁned as critical points of the super-symmetric energy functional. The Dirac-harmonic map is called R -trivial or uncoupled, if f is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps deﬁned on a compact domain are R -trivial. This raises the question whether all Dirac-harmonic maps with compact domain are R -trivial. We can apply similar arguments to the heat ﬂow for Dirac-harmonic maps, for which short-time existence and uniqueness was proven by Chen, Jost, Sun and Zhu [8] for ∂M ≠ ∅ and by Wittmann [18] for ∂M = ∅ . We show that this heat ﬂow is just an extension of the classical heat ﬂow for harmonic maps.

On triviality of dirac-harmonic maps

. Dirac-harmonic maps ( f,ϕ ) consist of a map f ∶ M → N and a twisted spinor ϕ ∈ Γ ( Σ M ⊗ f ∗ TN ) and they are deﬁned as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called uncoupled , if f is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps deﬁned on a closed domain are uncoupled.

Are all Dirac-harmonic maps uncoupled?

Let $$(\phi _\alpha , \psi _\alpha )$$
 be a sequence of $$\alpha $$
 -Dirac-harmonic maps from a Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy. If the target N is a sphere $$S^{K-1}$$
 , we show that the energy identity and necklessness hold during the interior blow-up process as $$\alpha \searrow 1$$
 for such a sequence .

Chaona Zhu

Papers

Energy identity and necklessness for $$\alpha $$ α -Dirac-harmonic maps into a sphere