#### Filter Results:

- Full text PDF available (42)

#### Publication Year

1970

2017

- This year (1)
- Last 5 years (10)
- Last 10 years (20)

#### Publication Type

#### Co-author

#### Journals and Conferences

#### Method

#### Organism

Learn More

- Mikhail G. Katz
- 1994

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifold S 2 Ã— S 2 admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case of S '~ Ã— S n, n > 3. Our point of departure is M. Gromov'sâ€¦ (More)

- Mikhail G. Katz
- 2005

P. Buser and P. Sarnak constructed Riemann surfaces whose systole behaves logarithmically in the genus. The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a newâ€¦ (More)

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologousâ€¦ (More)

- Mikhail G. Katz
- 2002

We prove an optimal systolic inequality for CAT(0) metrics on a genus 2 surface. We use a Voronoi cell technique, introduced by C. Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y = x âˆ’ x. Thus, among all CAT(0) metrics, the one with the best systolicâ€¦ (More)

In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By â€˜universalâ€™ we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assumptions, although somewhat arbitrary,â€¦ (More)

- Mikhail G. Katz, CHRISTINE LESCOP
- 2004

We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.

We generalize optimal inequalities of C. Loewner and M. Gromov, by proving lower bounds for the total volume in terms of the homotopy systole and the stable systole. Our main tool is the construction of an area-decreasing map to the Jacobi torus, streamlining and generalizing the construction of the first author in collaboration with D. Burago. It turns outâ€¦ (More)

- SCIENTIFIQUES DE Lâ€™Ã‰.N.S, IVAN BABENKO, Mikhail G. Katz
- 2004

In 1972, Marcel Berger defined a metric invariant that captures the 'size' of ^-dimensional homology of a Riemannian manifold. This invariant came to be called the fc-dimensional systole. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, inspired by M. Gromov's 1993 example,â€¦ (More)

- Mikhail G. Katz, YULI B. RUDYAK
- 2004

We show that the geometry of a Riemannian manifold (M,G) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M . It is denoted catsys(M), and defined in terms of the existence of systolicâ€¦ (More)

- Mikhail G. Katz
- 2004

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katokâ€™s lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromovâ€™s asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicativeâ€¦ (More)