arXiv research

Study finds new minimal surfaces in Schwarzschild space.

In a seminal paper published in $1968$, J. Simons proved that, for $n\leq 5$, the Euclidean (minimal) cone $CM$, built on a closed, oriented, minimal and non totally geodesic hypersurface $M^n$ of $\mathbb S^{n+1}$ is unstable. In this paper, we extend Simons' analysis to {\em warped} (minimal) cones built over a close…

We give a formula to calculate the indices of special (non-totally geodesic) minimal orbits of Hermann actions. Also, we give examples of such minimal orbits of Hermann actions and calculate their indices by using the formula.

Paper characterizes a special hypersurface in 5D sphere.

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones \cite{JLT} in dimension two. The s…

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler $\mathbb{S}^3 \times \mathbb{S}^3$ for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so called angle functions and that such Lagrangian submanifolds are closel…

Study calculates indices and nullities of focal manifolds in spheres.

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds…

Study calculates the renormalized area of catenoids in hyperbolic spaces.

Study of free boundary minimal Möbius bands in spherical caps.

Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.

A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k\big{(}\sqrt{\frac{k}{n}}\, \big{)}\times S^{n-k}\big{(}\sqrt{\frac{n-k}{n}}\, \big{)}$ gives the lowest value of area among all non-totally geodesic compact minimal hypersurfaces in the unit sphere $S^{n+1}(1)$. The presen…

The paper extends the Hopf differential concept to associative submanifolds in G2-manifolds.

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvatur…

New foliations found for critical surfaces of Hawking energy, resolving discrepancies.

In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and …

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the co…

It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equa…

The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.

Study on maximum principles for nonlinear equations on Riemannian manifolds.

The study examines the asymptotic behavior of cohomology groups of algebraic group subgroups.

The paper proves properties of triharmonic CMC hypersurfaces with limited curvature types.

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.

Study on homogeneous geodesics in sub-Riemannian geometry.

In non-compact manifolds, geodesic flowers exist.

Study on Mabuchi functional's convexity using ε-geodesics.

Geodesic graphs for special Finsler metrics on spheres are studied.

Characterizes visibility and geodesic loops in complex domains.

Conformal geodesics can't spiral in Riemannian manifolds.

Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.

New quasi-geodesics for Stiefel manifold simplify complex computations.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

Growth rates of geodesics on modular orbifolds are studied.

We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…

We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…

This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.

The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.

Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics betw…

No closed timelike geodesics in Kerr spacetimes, proving absence of closed causal geodesics.

Generic geodesic nets are dense in high-dimensional manifolds.

Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timeli…

The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…

Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.

Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.

The study examines geodesics and tight geodesics in surface curve complexes.

Theorem shows generic metrics yield non-degenerate geodesic nets.

New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…

Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.