Abstract
The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Note that \(ind _{f^*TN}(M)\) depends on the choice of a topological spin structure on M, but doesn’t depend on the Riemannian metrics on M and N. Moreover, \(ind _{g^*TN}(M)=ind _{f^*TN}(M)\) for any g in the homotopy class of f.
Parametrized on [0, 1]
For this chain of equations one has to be a little careful, since we argue with the curvature of \(F^*TN\), but F is only a \(C^1\)-mapping. However, all the expressions exist (e.g. in the sense that the exist in local coordinates) and all the equalities hold.
If \(m\not \equiv 3 \text { }(mod 4)\), then the spectrum of
is symmetric w.r.t. zero. This can be shown analogously to [13, Theorem 1.3.7 iv)].Here we use that \(p\ge 0\) and \(\int _M p(x,y,t)\,dV(y)=1\) for all \((x,t)\in M\times (0,\infty )\). Moreover, we use that there exists \(C>0\) s.t. \(\int _{0}^{t}\int _M|\nabla _x p(x,y,s)|\,dV(y)ds\le C \sqrt{t}\) for all \((x,t)\in M\times [0,1]\). The latter is not difficult to show. It follows directly from the construction of the heat kernel (see e.g. [5]). It is shown in detail in [11] or [30].
To show that \(W^{1,2,p}((0,T)\times M)\subset C^{0,1,\alpha }((0,T)\times M)\) for p large enough (the spaces are defined as in [27]) one needs the Sobolev embedding and interpolation theory.
Note that here \(T>0\) is just some T s.t. Theorem 1 holds. It does not need to be related to the T we constructed in the first step.
This can be seen as follows: we write \(u=u^i\). Since \(u\in C^{1,2,\alpha }((0,T)\times M)\subset C^{0,\frac{\alpha }{2}}((0,T);C^2(M))\) (c.f. [27]) we have in particular that \(u,\nabla u:(0,T)\rightarrow C^0(M)\) are \(\frac{\alpha }{2}\)-Hölder continuous. (In the case of \(\nabla u\) we write \(C^0(M)\) as target space shortly for \(\varGamma (TM)\) with the \(C^0\)-norm.) Hence \(u,\nabla u:(0,T)\rightarrow C^0(M)\) are uniformly continuous and can therefore be continuously extended to \(u,\nabla u:[0,T]\rightarrow C^0(M)\). Hence \(u(t,.)\rightarrow u_0\) in \(C^0(M)\) as \(t\rightarrow 0\) and there exists a vector field \(V\in \varGamma (TM)\) s.t. \(\nabla u(t,.)\rightarrow V\) in \(C^0(M)\) as \(t\rightarrow 0\). We show \(V=\nabla u_0\). To that end, notice that for every \(X\in \varGamma (TM)\) we have
$$\begin{aligned} \int _M\langle \nabla u(t,.),X\rangle =-\int _Mu(t,.)div (X)\xrightarrow {t\rightarrow 0} -\int _M u_0div (X)=\int _M\langle \nabla u_0,X\rangle , \end{aligned}$$and
$$\begin{aligned} \int _M\langle \nabla u(t,.),X\rangle \xrightarrow {t\rightarrow 0}\int _M\langle V,X\rangle . \end{aligned}$$Since
and 
, we can assume w.l.o.g. that \(\psi ^1_{T_0}=\psi ^2_{T_0}\). Otherwise we replace \(\psi ^2\) by \(\psi ^2h\), where \(h\in \mathbb {K}\) has unit length with \(\psi ^1_{T_0}=\psi ^2_{T_0}h\).
is symmetric w.r.t. zero. This can be shown analogously to [
and 
, we can assume w.l.o.g. that References
Ammann, B.: A Variational Problem in Conformal Spin Geometry. Universität Hamburg, Habilitationsschrift (2003)
Ammann, B., Ginoux, N.: Examples of Dirac-harmonic maps after Jost-Mo-Zhu. http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/ammann/preprints/diracharm/diracharm_JostMoZhu09.pdf (2009)
Ammann, B., Ginoux, N.: Dirac-harmonic maps from index theory. Calc. Var. Partial Differ. Equ. 47(3–4), 739–762 (2013)
Branding, V.: The evolution equations for Dirac-harmonic Maps. Ph.D. thesis, Universität Potsdam (2013)
Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, Cambridge (1984)
Chen, Q., Jost, J., Li, J., Wang, G.: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 251(1), 61–84 (2005)
Chen, Q., Jost, J., Li, J., Wang, G.: Dirac-harmonic maps. Math. Z. 254(2), 409–432 (2006)
Chen, Q., Jost, J., Sun, L., Zhu, M.: Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem. To appear in J. Eur. Math. Soc
Chen, Q., Jost, J., Wang, G.: The maximum principle and the Dirichlet problem for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 47(1–2), 87–116 (2013)
Chen, Q., Jost, J., Wang, G., Zhu, M.: The boundary value problem for Dirac-harmonic maps. J. Eur. Math. Soc. 15(3), 997–1031 (2013)
Dietert, H., Moore, K., Rockstroh, P., Shaw, G.: Heat Flow Methods in Geometry and Topology. https://cmouhot.files.wordpress.com/1900/10/harmonicmaps.pdf
Friedrich, T.: Dirac Operators in Riemannian Geometry. Graduate Studies in Mathematics. AMS, Providence (2000)
Ginoux, N.: The Dirac Spectrum, Volume 1976 of Lecture Notes in Mathematics, vol. 1976. Springer, Berlin (2009)
Hermann, A.: Dirac eigenspinors for generic metrics. PhD thesis, Universität Regensburg (2012)
Hijazi, O.: Spectral properties of the Dirac operator and geometrical structures. In: Proceedings of the Summer School on Geometric Methods in Quantum Field Theory, Villa de Leyva, Colombia, 12–30 July 2001, World Scientific (1999)
Jost, J.: Riemannian Geometry and Geometric Analysis, sixth edn. Springer, Berlin (2011)
Jost, J., Liu, L., Zhu, M.: Existence of solutions of a mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor. MPI MIS Preprint: 35/2017
Jost, J., Liu, L., Zhu, M.: A global weak solution of the Dirac-harmonic map flow. To appear in Ann. Inst. H. Poincare Anal. Non Lineaire (2017). https://doi.org/10.1016/j.anihpc.2017.01.002
Jost, J., Mo, X., Zhu, M.: Some explicit constructions of Dirac-harmonic maps. J. Geom. Phys. 59(11), 1512–1527 (2009)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (2013). (Reprint of the 1980 Edition)
Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)
Lin, F., Wang, C.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific, Singapore (2008)
Liu, L.: No neck for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 52(1–2), 1–15 (2015)
Nowaczyk, N.: Dirac Eigenvalues of higher Multiplicity. Ph.D. thesis, Universität Regensburg (2014)
Sharp, B., Zhu, M.: Regularity at the free boundary for Dirac-harmonic maps from surfaces. Calc. Var. Partial Differ. Equ. 55(2), 27 (2016)
Smith, P.D., Yang, D.: Removing point singularities of Riemannian manifolds. Trans. Am. Math. Soc. 333(1), 203–219 (1992)
Tapio, B.: Generalized Lagrangian mean curvature flow in almost Calabi–Yau manifolds. PhD thesis, University of Oxford (2011)
Wang, C., Xu, D.: Regularity of Dirac-harmonic maps. Int. Math. Res. Not. 20, 3759–3792 (2009)
Warner, F.: Extension of the Rauch comparison theorem to submanifolds. Trans. Am. Math. Soc. 122(2), 341–356 (1966)
Wittmann, J.: Ph.D. thesis. In prepartion
Zhao, L.: Energy identities for Dirac-harmonic maps. Calc. Var. Part. Differ. Equ. 28(1), 121–138 (2007)
Zhu, M.: Dirac-harmonic maps from degenerating spin surfaces. I. The Neveu-Schwarz case. Calc. Var. Partial Differ. Equ. 35(2), 169–189 (2009)
Zhu, M.: Regularity for weakly Dirac-harmonic maps to hypersurfaces. Ann. Global Anal. Geom. 35(4), 405–412 (2009)
Acknowledgements
I would like to thank Bernd Ammann and Helmut Abels for their ongoing support and many fruitful discussions. I am also grateful to Ulrich Bunke for his valuable input. My work was supported by the DFG Graduiertenkolleg GRK 1692 “Curvature, Cycles, and Cohomology”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jost.
Rights and permissions
About this article
Cite this article
Wittmann, J. Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds. Calc. Var. 56, 169 (2017). https://doi.org/10.1007/s00526-017-1270-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1270-1