Maps from Riemannian manifolds into non-degenerate Euclidean cones. (English) Zbl 1205.53006

Summary: Let \(M\) be a connected, non-compact \(m\)-dimensional Riemannian manifold. We consider smooth maps \(\varphi: M \rightarrow \mathbb{R}^n\) with images inside a non-degenerate cone. Under quite general assumptions on \(M\), we provide a lower bound for the width of the cone in terms of the energy and the tension of \(\varphi\) and a metric parameter. We recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case \(\varphi\) is an isometric immersion, we also show that, if \(M\) is sufficiently well-behaved and has non-positive sectional curvature, \(\varphi(M)\) cannot be contained in a non-degenerate cone of \(\mathbb{R}^{2m-1}\).


53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
35B50 Maximum principles in context of PDEs
53C43 Differential geometric aspects of harmonic maps
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[1] Atsuji, A.: Remarks on harmonic maps into a cone from a stochastically complete manifold. Proc. Japan Acad. Ser.A Math. Sci. 75 (1999), no. 7, 105-108. · Zbl 0960.58019
[2] Alías, L.J. and García-Martínez, S.C.: On the scalar curvature of constant mean curvature hypersurfaces in space forms. J. Math. Anal. Appl. 363 (2010), no. 2, 579-587. · Zbl 1182.53052
[3] Alías, L.J., Pacelli Bessa, G. and Dajczer, M.: The mean curvature of cylindrically bounded submanifolds. Math. Ann. 345 (2009), no. 2, 367-376. · Zbl 1200.53050
[4] Baikoussis, C. and Kouforgiorgos, T.: Harmonic maps into a cone. Arch. Math. (Basel) 40 (1983), no. 4, 372-386.
[5] Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics 115 . Academic Press, Orlando, FL, 1984. · Zbl 0551.53001
[6] Chern, S.S. and Kuiper, N.H.: Some theorems on the isometric imbedding of compact Riemannian manifolds in Euclidean space. Annals of Math. (2) 56 (1952), 422-430. JSTOR: · Zbl 0052.17601
[7] Dajczer, M.: Submanifolds and Isometric immersions. Mathematical Lecture Series 13 . Publish or Perish, Houston, TX, 1990. · Zbl 0705.53003
[8] Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36 (1999), no. 2, 135-249. · Zbl 0927.58019
[9] Greene, R. and Wu, H.H.: Function theory on manifolds which possess a pole. Lecture Notes on Mathematics 699 . Springer-Verlag, Berlin, 1979. · Zbl 0414.53043
[10] Hoffman, D. and Meeks, W.: The strong halfspace theorem for minimal surfaces. Invent. Math. 101 (1990), no. 2, 373-377. · Zbl 0722.53054
[11] Jorge, L. and Koutroufiotis, D.: An estimate for the curvature of bounded submanifolds. Amer. J. Math. 103 (1981), no. 4, 711-725. JSTOR: · Zbl 0472.53055
[12] Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19 (1967), 205-214. · Zbl 0154.21501
[13] Pigola, S., Rigoli, M. and Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Memoirs of the Am. Math. Soc. 174 (2005). · Zbl 1075.58017
[14] Pigola, S., Rigoli, M. and Setti, A.G.: A remark on the maximum principle and stochastic completeness. Proc. Amer. Math. Soc. 131 (2003), no. 4, 1283-1288. JSTOR: · Zbl 1015.58007
[15] Pigola, S., Rigoli, M. and Setti, A.G.: Volume growth, “a priori” estimates and geometric applications. Geom. Funct. Anal. 13 (2003), no. 6, 1302-1328. · Zbl 1065.53037
[16] Pigola, S., Rigoli, M. and Setti, A.G.: Maximum principles at infinity on Riemannian manifolds: an overview. Mat. Contemp. 31 (2006), 81-128. · Zbl 1145.58009
[17] Ranjbar-Motlagh, A.: On harmonic maps from stochastically complete manifolds. Arch. Math. (Basel) 92 (2009), no. 6, 637-644. · Zbl 1180.58011
[18] Rigoli, M., Salvatori, M. and Vignati, M.: Some remarks on the weak maximum principle. Rev. Mat. Iberoamericana 21 (2005), no. 2, 459-481. · Zbl 1110.58022
[19] Sturm, K.T.: Analysis on local Dirichlet spaces I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine Angew. Math. 456 (1994), 173-196. · Zbl 0806.53041
[20] Tompkins, C.: Isometric embedding of flat manifolds in Euclidean space. Duke Math. J. 5 (1939), no. 1, 58-61. · Zbl 0020.39701
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