## Duke Mathematical Journal

### Families of short cycles on Riemannian surfaces

Yevgeny Liokumovich

#### Abstract

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a nontrivial element of the kth homology group of the space of cycles and such that the mass of each cycle is bounded above by $C\max\{\sqrt{k},\allowbreak \sqrt{g}\}\sqrt{\operatorname {Area}(M)}$. This result is optimal up to a multiplicative constant.

#### Article information

Source
Duke Math. J., Volume 165, Number 7 (2016), 1363-1379.

Dates
Revised: 16 August 2015
First available in Project Euclid: 5 February 2016

https://projecteuclid.org/euclid.dmj/1454683425

Digital Object Identifier
doi:10.1215/00127094-3450208

Mathematical Reviews number (MathSciNet)
MR3498868

Zentralblatt MATH identifier
1341.53072

#### Citation

Liokumovich, Yevgeny. Families of short cycles on Riemannian surfaces. Duke Math. J. 165 (2016), no. 7, 1363--1379. doi:10.1215/00127094-3450208. https://projecteuclid.org/euclid.dmj/1454683425

#### References

• [1] F. J. Almgren, Jr, The homotopy groups of the integral cycle groups, Topology 1 (1962), 257–299.
• [2] F. Balacheff and S. Sabourau, Diastolic and isoperimetric inequalities on surfaces, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 579–605.
• [3] R. Brooks, The spectral geometry of a tower of coverings, J. Differ. Geom. 23 (1986), 97–107.
• [4] P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progr. in Math. 106, Springer, New York, 1992.
• [5] B. Colbois and D. Maerten, Eigenvalues estimate for the Neumann problem of bounded domain, J. Geom. Anal. 18 (2008), 1022–1032.
• [6] H. Federer, Geometric Measure Theory, Springer, New York, 1969.
• [7] P. Glynn-Adey and Y. Liokumovich, Width, Ricci curvature and minimal hypersurfaces, preprint, arXiv:1408.3656v4 [math.DG].
• [8] A. Grigor’yan, Y. Netrusov, and S.-T. Yau, “Eigenvalues of elliptic operators and geometric applications” in Surveys in Differential Geometry, Surv. Differ. Geom. IX, Int. Press, Somerville, Mass., 2004, 147–217.
• [9] M. Gromov, “Dimension, non-linear spectra and width” in Geometric Aspects of Functional Analysis, Springer, New York, 1988, 132–184.
• [10] M. Gromov, “Metric invariants of Kähler manifolds” in Differential Geometry and Topology (Alghero, 1992), World Sci. Publ., River Edge, N.J., 1993, 90–116.
• [11] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178–215.
• [12] M. Gromov, Singularities, expanders and topology of maps, I: Homology versus volume in the spaces of cycles, Geom. Funct. Anal. 19 (2009), 743–841.
• [13] L. Guth, The width-volume inequality, Geom. Funct. Anal. 17 (2007), 1139–1179.
• [14] L. Guth, Minimax problems related to cup powers and Steenrod squares, Geom. Funct. Anal. 18 (2009), 1917–1987.
• [15] A. Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, J. Funct. Anal. 261 (2011), 3419–3436.
• [16] J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A 270 (1970), 1645–1648.
• [17] J. Jenkins, Univalent Functions and Conformal Mapping, Ergeb. Math. Grenzgeb., Springer, Berlin, 1958.
• [18] N. Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differ. Geom. 37 (1993), 73–93.
• [19] P. Li and S.-T. Yau, A new conformal invariant and its applications to the Wilmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269–292.
• [20] Y. Liokumovich, Slicing a 2-sphere, J. Topol. Anal. 6 (2014), 573–590.
• [21] Y. Liokumovich, A. Nabutovsky, and R. Rotman, Contracting the boundary of a Riemannian 2-disc, preprint, arXiv:1205.5474v2 [math.DG].
• [22] F. C. Marques and A. Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, preprint, arXiv:1311.6501v1 [math.DG].
• [23] P. Yang and S. T. Yau, Eigenvalues of the laplacian of compact riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1980), 55–63.
• [24] S.-T. Yau, Isoperimetric constants and the first eigenvalue of a compact riemannian manifold, Ann. Sci. Éc. Norm. Supér. 8 (1975), 487–507.