Abstract
We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold \(\Sigma \) of a Riemannian manifold \(\mathcal {M}\). We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of multiple valued sections of the normal bundle of \(\Sigma \) in \(\mathcal {M}\), and we study existence and regularity of such minimizers. Finally, we prove that any Q-valued Jacobi field can be written as the superposition of Q classical Jacobi fields everywhere except for a relatively closed singular set having codimension at least two in the domain.
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Notes
Observe that \(F \circ u\) is a well defined function \(\Omega \rightarrow \mathbb {R}^{q}\), because F is, by hypothesis, a well defined map on the quotient \(\mathcal {A}_{Q}(\mathbb {R}^d) = (\mathbb {R}^d)^{Q}/\mathcal {P}_{Q}\).
That is, \(\vec {\xi }(x)\) is a continuous unit m-vector field on \(\Sigma \) with \((\xi _i)_{i=1}^{m}\) an orthonormal frame of the tangent bundle \(\mathcal {T}\Sigma \).
Observe that this convention is coherent with the use of \(\llbracket p \rrbracket \), \(p \in \mathbb {R}^d\), to denote the Dirac delta \(\delta _{p}\), considered as a 0-dimensional current in \(\mathbb {R}^d\).
Here, the Sobolev space \(W^{1,p}(\Omega )\) is classically defined as the completion of \(C^{1}(\Omega )\) with respect to the \(W^{1,p}\)-norm
$$\begin{aligned} \Vert f \Vert _{W^{1,p}(\Omega )}^{p} := \int _{\Omega } \left( |f(x)|^{p} + |Df(x)|^{p} \right) \, \mathrm{d}\mathcal {H}^{m}(x) \end{aligned}$$for \(1 \le p < \infty \) and
$$\begin{aligned} \Vert f \Vert _{W^{1,\infty }(\Omega )} := \mathop {{{\,\mathrm{ess\,sup}\,}}}\limits _{\Omega } \left( |f(x)| + |Df(x)| \right) . \end{aligned}$$Observe that if \(N|_{\partial \Omega } = Q \llbracket 0 \rrbracket \), then the null Q-field \(N_{0} \equiv Q \llbracket 0 \rrbracket \) is a competitor, whence \(\mathrm {Jac}(N,\Omega ) \le \mathrm {Jac}(N_{0},\Omega ) = 0\) if N is a minimizer.
Recall that the existence of such a map f is guaranteed by Theorem 1.25.
Observe that the inequality
$$\begin{aligned} \mathrm {Dir}(\mathscr {N}_{p}, B_{\rho }) \le D_{\rho } \end{aligned}$$is guaranteed for every \(\rho \) because the Dirichlet functional is lower semi-continuous with respect to weak convergence in \(W^{1,2}\).
References
Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95, 417–491 (1972)
Allard, W.K., Almgren Jr., F.J.: On the radial behavior of minimal surfaces and the uniqueness of their tangent cones. Ann. Math. (2) 113, 215–265 (1981). https://doi.org/10.2307/2006984
Almgren, Jr., F.J.: Almgren’s Big Regularity Paper: \(Q\)-valued Functions Minimizing Dirichlet’s Integral and the Regularity of Area-Minimizing Rectifiable Currents Up to Codimension 2. World Scientific Monograph Series in Mathematics, vol. 1. World Scientific Publishing Co., Inc., River Edge, NJ (2000) (with a preface by Jean E. Taylor and Vladimir Scheffer)
Ambrosio, L.: Metric space valued functions of bounded variation. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 17, 439–478 (1990). http://www.numdam.org/item?id=ASNSP_1990_4_17_3_439_0
Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969). https://doi.org/10.1007/BF01404309
do Carmo, M.P.: Riemannian Geometry, Mathematics: Theory & Applications (Translated from the second Portuguese edition by Francis Flaherty). Birkhäuser Boston, Inc., Boston, MA (1992). https://doi.org/10.1007/978-1-4757-2201-7
De Lellis, C.: Errata to “Q-valued functions revisited”. http://www.math.uzh.ch/fileadmin/user/delellis/publikation/Errata_memo.pdf (2013). Accessed Oct 2016
De Lellis, C.: The regularity of minimal surfaces in higher codimension. In: Jerison, D., Kisin, M., Seidel, P., Stanley, R., Yau, H.-T., Yau, S.-T. (eds.) Current Developments in Mathematics 2014, pp. 153–229. International Press, Somerville, MA (2016)
De Lellis, C.: The size of the singular set of area-minimizing currents. In: Cao, H.-D., Yau, S.-T. (eds.) Surveys in Differential Geometry 2016. Advances in Geometry and Mathematical Physics. Surveys in Differential Geometry, vol. 21, pp. 1–83. International Press, Somerville, MA (2016)
De Lellis, C., Focardi, M., Spadaro, E.N.: Lower semicontinuous functionals for Almgren’s multiple valued functions. Ann. Acad. Sci. Fenn. Math. 36, 393–410 (2011). https://doi.org/10.5186/aasfm.2011.3626
De Lellis, C., Hirsch, J., Marchese, A., Stuvard, S.: Regularity of area minimizing currents modulo \(p\) (forthcoming) (2019)
De Lellis, C., Marchese, A., Spadaro, E., Valtorta, D.: Rectifiability and upper Minkowski bounds for singularities of harmonic \(Q\)-valued maps. Comment. Math. Helv. 93, 737–779 (2018). https://doi.org/10.4171/CMH/449
De Lellis, C., Spadaro, E.: Regularity of area minimizing currents I: gradient \(L^p\) estimates. Geom. Funct. Anal. 24, 1831–1884 (2014). https://doi.org/10.1007/s00039-014-0306-3
De Lellis, C., Spadaro, E.: Multiple valued functions and integral currents. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XIV, 1239–1269 (2015)
De Lellis, C., Spadaro, E.: Regularity of area minimizing currents II: center manifold. Ann. Math. (2) 183, 499–575 (2016). https://doi.org/10.4007/annals.2016.183.2.2
De Lellis, C., Spadaro, E.: Regularity of area minimizing currents III: blow-up. Ann. Math. (2) 183, 577–617 (2016). https://doi.org/10.4007/annals.2016.183.2.3
De Lellis, C., Spadaro, E., Spolaor, L.: Regularity theory for 2-dimensional almost minimal currents II: branched center manifold. Ann. PDE 3(2), Art. 18 (2017). https://doi.org/10.1007/s40818-017-0035-7
De Lellis, C., Spadaro, E., Spolaor, L.: Regularity theory for \(2\)-dimensional almost minimal currents I: Lipschitz approximation. Trans. Am. Math. Soc. 370, 1783–1801 (2018). https://doi.org/10.1090/tran/6995
De Lellis, C., Spadaro, E., Spolaor, L.: Regularity theory for 2-dimensional almost minimal currents III: blowup. arXiv:1508.05510 (to appear on J. Differ. Geom) (2019)
De Lellis, C., Spadaro, E.N.: \(Q\)-valued functions revisited. Mem. Am. Math. Soc. 211, vi+79 (2011). https://doi.org/10.1090/S0065-9266-10-00607-1
Federer, H.: Some theorems on integral currents. Trans. Am. Math. Soc. 117, 43–67 (1965)
Federer, H.: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)
Giaquinta, M., Modica, G., Souček, J.R.: Cartesian Currents in the Calculus of Variations. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 37. Springer, Berlin (1998). https://doi.org/10.1007/978-3-662-06218-0
Hang, F.: On the weak limits of smooth maps for the Dirichlet energy between manifolds. Commun. Anal. Geom. 13, 929–938 (2005). https://doi.org/10.4310/CAG.2005.v13.n5.a4
Hirsch, J.: Boundary regularity of Dirichlet minimizing \(Q\)-valued functions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16, 1353–1407 (2016)
Hirsch, J.: Partial Hölder continuity for \(Q\)-valued energy minimizing maps. Commun. Partial Differ. Equ. 41, 1347–1378 (2016). https://doi.org/10.1080/03605302.2016.1204313
Hirsch, J., Stuvard, S., Valtorta, D.: Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps. Trans. Am. Math. Soc. 371, 4303–4352 (2019). https://doi.org/10.1090/tran/7595
Krantz, S.G., Parks, H.R.: Geometric Integration Theory, Cornerstones. Birkhäuser Boston Inc., Boston (2008). https://doi.org/10.1007/978-0-8176-4679-0
Krummel, B., Wickramasekera, N.: Fine properties of branch point singularities: two-valued harmonic functions. arXiv:1311.0923 (2013)
Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997). https://doi.org/10.1007/b98852
Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J. 37, 349–367 (1988). https://doi.org/10.1512/iumj.1988.37.37017
Morgan, F.: On the singular structure of two-dimensional area minimizing surfaces in \({ R}^{n}\). Math. Ann. 261, 101–110 (1982). https://doi.org/10.1007/BF01456413
Moser, R.: Partial Regularity for Harmonic Maps and Related Problems. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005). https://doi.org/10.1142/9789812701312
Reshetnyak, Y.G.: Sobolev classes of functions with values in a metric space. Sib. Mat. Zhurnal 38, 567–583 (1997). https://doi.org/10.1007/BF02683844
Reshetnyak, Y.G.: Sobolev classes of functions with values in a metric space. II. Sib. Mat. Zhurnal 45, 855–870 (2004). https://doi.org/10.1023/B:SIMJ.0000035834.03736.b6
Reshetnyak, Y.G.: On the theory of Sobolev classes of functions with values in a metric space. Sib. Mat. Zhurnal 47, 146–168 (2006). https://doi.org/10.1007/s11202-006-0013-x
Simon, L.: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. (2) 118, 525–571 (1983). https://doi.org/10.2307/2006981
Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983)
Simon, L.: Uniqueness of some cylindrical tangent cones. Commun. Anal. Geom. 2, 1–33 (1994). https://doi.org/10.4310/CAG.1994.v2.n1.a1
Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1996). https://doi.org/10.1007/978-3-0348-9193-6. (based on lecture notes by Norbert Hungerbühler)
Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968)
Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute. In: Brothers, J.E. (ed.) Geometric Measure Theory and the Calculus of Variations (Arcata, Calif., 1984). Proceedings of Symposia in Pure Mathematics, vol. 44, pp. 441–464. American Mathematical Society, Providence, RI (1986). https://doi.org/10.1090/pspum/044/840292
Stuvard, S.: Multiple valued sections of vector bundles: the reparametrization theorem for \(Q\)-valued functions revisited. arXiv:1705.00054 (2017)
White, B.: Tangent cones to two-dimensional area-minimizing integral currents are unique. Duke Math. J. 50, 143–160 (1983). https://doi.org/10.1215/S0012-7094-83-05005-6
White, B.: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps. Acta Math. 160, 1–17 (1988). https://doi.org/10.1007/BF02392271
Acknowledgements
The author is warmly thankful to Camillo De Lellis for suggesting him to study this problem, and for his precious guidance and support; and to Guido De Philippis, Francesco Ghiraldin, and Luca Spolaor for several useful discussions. The author also thanks the anonymous reviewer for carefully reading the manuscript, and for his/her valuable comments. The research of S.S. has been supported by the ERC Grant Agreement RAM (Regularity for Area Minimizing currents), ERC 306247.
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Communicated by L. Ambrosio.
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