| 1. |
- Tkachev, Vladimir
(författare)
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A theorem on the clearance radius for minimal surfaces
- 1996
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Ingår i: Mathematical notes of the Academy of Sciences of the USSR. - : Springer Berlin/Heidelberg. - 0001-4346 .- 1573-8876. ; 59:5-6, s. 657-660
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Tidskriftsartikel (refereegranskat)abstract
- A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.
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| 2. |
- Tkachev, Vladimir, et al.
(författare)
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Denjoy-Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces
- 1996
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Ingår i: Communications in analysis and geometry. - : International Press of Boston. - 1019-8385 .- 1944-9992. ; 4:4, s. 547-587
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Tidskriftsartikel (refereegranskat)abstract
- We extend the well-known Denjoy-Ahlfors theorem about the number of different asymptotic tracts of a holomorphic function to subharmonic functions on arbitrary Riemannian manifolds. We obtain some versions of the Liouville theorem for Aa-harmonic functions without the geodesic completeness requirement on a manifold. Moreover, the upper estimate of the topological index of the height function of a minimal surface in Rn has been established and, as a consequence, a new prove of the Bernstein's theorem has been derived. Other applications to the theory of minimal surfaces are also discussed.
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| 3. |
- Tkachev, Vladimir
(författare)
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External geometry of p-minimal surfaces
- 1997
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Ingår i: Geometry from Pacific Rim. - Berlin, New York : Walter de Gruyter. - 9783110908961 ; , s. 363-376
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Konferensbidrag (refereegranskat)abstract
- We introduce a new class of p-minimal surfaces. We shaw that the Gauss map of a two-dimensional p-minimal surface in R3 is quasiconformal . A counterpart to Bernstein's celebrated result about entire solutions of the minimal surface equation is obtained. A study of tubular p-minimal hypersurfaces is included.
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| 4. |
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| 5. |
- Tkachev, Vladimir
(författare)
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Minimal tubes of finite integral curvature
- 1998
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Ingår i: Siberian mathematical journal. - : Springer Berlin/Heidelberg. - 0037-4466 .- 1573-9260. ; 39:1, s. 159-167
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Tidskriftsartikel (refereegranskat)abstract
- The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M.
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| 6. |
- Tkachev, Vladimir, et al.
(författare)
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On the Gauss map of embedded minimal tubes
- 1999
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Ingår i: Note di Matematica. - University of Salento : SIBA. - 1123-2536 .- 1590-0932. ; 19:1, s. 7-17
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Tidskriftsartikel (refereegranskat)abstract
- A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alpha(M) between the axis and the flow vector of M. We prove that the diameter of the Gauss image of M is at least 2alpha(M). As a consequence we derive an estimate on the length of a two-dimensional minimal tube M in terms of alpha(\M) and the total Gaussian curvature of M.
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| 7. |
- Andersson, Lars, et al.
(författare)
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Boundary and lens rigidity of Lorentzian surfaces
- 1996
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Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 348, s. 2307-2329
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Tidskriftsartikel (refereegranskat)abstract
- Let g be a Lorentzian metric on the plane ℝ2 that agrees with the standard metric g0 = -dx2 + dy2 outside a compact set and so that there are no conjugate points along any time-like geodesic of (ℝ2, g). Then (ℝ2, g) and (ℝ2, g0) are isometric. Further, if (M*, g*) and (M*, p*) are two dimensional compact time oriented Lorentzian manifolds with space-like boundaries and so that all time-like geodesies of (M, g) maximize the distances between their points and (M, g) and (M*, g*) are "boundary isometric", then there is a conformal diffeomorphism between (M, g) and (M*, g*) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.
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| 8. |
- Gosson, Maurice de
(författare)
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On the classical and quantum evolution of Lagrangian half-forms in phase space
- 1999
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Ingår i: ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE. - PARIS : GAUTHIER-VILLARS/EDITIONS ELSEVIER. - 0246-0211. ; , s. 547-573
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Tidskriftsartikel (refereegranskat)abstract
- The local expressions of a Lagrangian half-form on a quantized Lagrangian submanifold of phase space are the wavefunctions of quantum mechanics. We show that one recovers Maslov's asymptotic formula for the solutions to Schrodinger's equation if one transports these half-forms by the flow associated with a Hamiltonian H. We then consider the case when the Hamiltonian flow is replaced by the flow associated with the Bohmian, and are led to the conclusion that the use of Lagrangian half-forms leads to a quantum mechanics on phase space. (C) Elsevier, Paris.
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| 9. |
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| 10. |
- Gudmundsson, Sigmundur
(författare)
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Harmonic morphisms as sphere bundles over compact Riemann surfaces
- 1997
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Ingår i: International Journal of Mathematics. - 0129-167X. ; 8:7, s. 935-942
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Tidskriftsartikel (refereegranskat)abstract
- We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.
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