Polar varieties and bipolar surfaces of minimal surfaces in the n-sphere

For a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson’s polar variety and bipolar surface.

An efficient protocol for chromosome isolation from sterlet (A. ruthenus) embryos and larvae

In the present study, the development of an efficient and feasible protocol for chromosome preparation from sterlet (A. ruthenus) embryos and larvae was carried out. In the established protocol, the mean efficiency of chromosome extraction ranged from 70 to 100%. The average number of recorded metaphases per slide was between 9 to 15. In general, the most satisfactory results were obtained for embryos at 6 dpf and larvae at the age of up to 7 dph. In the 24 dpf group, chromosome isolation was possible without immersion in spindle poison, however; in successive developmental stages, the minimal immersion time exceeded 1.5 hours, regardless of chorionation. Immersion for 14 hours did not compromise the efficacy of chromosome isolation. In the current study, successful chromosome isolation was determined mainly by hypotonization conditions. Younger developmental stages generally require the shortest hypotonization times, whereas older larvae require longer hypotonization times. The optimal hypotonization period is 5-15 minutes for embryos at 24 dpf, 40 minutes for embryos at 4dpf, and 50-60 minutes for fish at 6 dpf-7 dph. The only exception was the 24 hpf group where only blastula cells were used. An additional overnight fixation step significantly enhanced chromosome quality and supported chromosome counting especially in the 24 dpf group. The quality and quantity of chromosome slides were also significantly determined by tissue type, and the slides prepared from heads (gill cells) produced the best results.

Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{C}^{3}} (i.e. {u_{1}} has vanishing flux). If furthermore u is nonflat, then {u_{1}} can be chosen to have any prescribed flux and to be complete.

Colliding vowel systems in Andean Spanish

The acquisition of the Spanish 5-vowel system by speakers of the 3-vowel language Quechua (/I/-/a/-/ʊ/) seldom results in accurate approximation to Spanish vowel spaces when learning takes place informally in post-adolescence. The present study offers data from a minimal immersion environment in northern Ecuador. In a context in which few cues point to the existence of mid-high vocalic oppositions in Spanish (e.g. no literacy, no corrective feedback, almost no viable minimal pairs), these speakers reliably distinguish only three Spanish vowels in production. These Quechua-dominant bilinguals have amorphous front and back vowel spaces considerably broader than those defining Quechua /I/ and /ʊ/, but with no bimodal clustering. Left relatively unfettered, the hybrid system may contribute to an understanding of the relationship between vowel inventory and vowel space topology.

Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface

Let L be an oriented Lagrangian submanifold in an n-dimensional Kähler manifold M. Let u: D → M be a minimal immersion from a disk D with u(𝜕D) ⊂ L such that u(D) meets L orthogonally along u(𝜕D). Then the real dimension of the space of admissible holomorphic variations is at least n + μ(E, F), where μ(E, F) is a boundary Maslov index; the minimal disk is holomorphic if there exist n admissible holomorphic variations that are linearly independent over ℝ at some point p ∈ 𝜕D; if M = ℂPn and u intersects L positively, then u is holomorphic if it is stable, and its Morse index is at least n + μ(E, F) if u is unstable.

GROWTH ESTIMATES FOR WARPING FUNCTIONS AND THEIR GEOMETRIC APPLICATIONS

By applying Wei, Li and Wu's notion (given in ‘Generalizations of the uniformization theorem and Bochner's method in p-harmonic geometry’, Comm. Math. Anal. Conf., vol. 01, 2008, pp. 46–68) and method (given in ‘Sharp estimates on -harmonic functions with applications in biharmonic maps, preprint) and by modifying the proof of a general inequality given by Chen in ‘On isometric minimal immersion from warped products into space forms’ (Proc. Edinb. Math. Soc., vol. 45, 2002, pp. 579–587), we establish some simple relations between geometric estimates (the mean curvature of an isometric immersion of a warped product and sectional curvatures of an ambient m-manifold $\tilde M^m_c$ bounded from above by a non-positive number c) and analytic estimates (the growth of the warping function). We find a dichotomy between constancy and ‘infinity’ of the warping functions on complete non-compact Riemannian manifolds for an appropriate isometric immersion. Several applications of our growth estimates are also presented. In particular, we prove that if f is an Lq function on a complete non-compact Riemannian manifold N1 for some q > 1, then for any Riemannian manifold N2 the warped product N1 ×fN2 does not admit a minimal immersion into any non-positively curved Riemannian manifold. We also show that both the geometric curvature estimates and the analytic function growth estimates in this paper are sharp.

STALLINGS FOLDINGS AND SUBGROUPS OF AMALGAMS OF FINITE GROUPS

Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

ON ISOMETRIC MINIMAL IMMERSIONS FROM WARPED PRODUCTS INTO REAL SPACE FORMS

We establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25