arXiv research

Invariants of Finsler manifolds are expressed using moving planes and Jacobi curves.

Method of moving planes used for proving symmetry in PDEs and geometric analysis.

New moving plane method for varifolds promotes smoothness from boundary to interior.

Paper introduces clock moves for plane graphs and proves Alexander polynomial properties.

New moves for singular knots identified and described.

Study of curves in affine space with moving frames and invariants.

A {\em blink} is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, I show that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a…

The paper introduces a new Markov chain sampler for knot diagrams.

Extends symmetry and rigidity to surfaces with soap film-like singularities.

A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.

Cactus doodles are geometric objects derived from cactus groups.

Proves rotational symmetry for Serrin-type problems in doubly connected domains.

Study shows curves converge to traveling waves under specific conditions.

In this paper, by the method of moving planes, we prove the symmetry result which says that classical solutions of Monge-Ampere system in the whole plane are symmetric about some point. Our system under consideration comes from the differential geometry problem.

In 2001, Oestlund conjectured that Reidemeister moves 1 and 3 are sufficient to describe a homotopy from any generic immersion from the circle into the plane to the standard embedding of the circle. We show that this conjecture is false.

Geometric triangulations can be transformed by bistellar moves.

Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.

Paper defines curvature equivalence for Legendre curves in a plane.

Sharp stability estimates for hypersurfaces in hyperbolic space.

New method studies moving points on curves using rotating frames.

Transforming curves with crossings requires a number of moves proportional to n^(3/2).

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

No theta-shaped networks shrink similarly under curvature.

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-paramete…

The paper studies how the crossing number of graphs changes with a specific transformation called ΔY-move.

A new formula detects differences between counterexamples and standard embeddings of circles.

Characterizes spherical and plane curves using RM frames.

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $Ω(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $Δ$Y transformations are required in the worst case to reduce an $n$-vertex plane graph to a single vertex or edge. The lower bound is achieved by any planar g…

The paper proves a conjecture about mean curvature flows in higher dimensions.

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by…

The paper discusses Nirenberg's work on geometric problems and his personality.

Study on diagrams of links with triple-crossings and their properties.

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.

New method proves hypersurface symmetry under weaker conditions.

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

The paper improves proximity estimates for hypersurfaces with almost constant curvature in space forms.

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…

Paper analyzes Libor rates using information theory to detect abnormal movements around the financial crisis.

Study motion planning for points avoiding obstacles in a plane.

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $-Δu=K(x)e^u, in R^2$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite to…

Paper extends method of presenting surface-links to immersed surface-links.

Study of nonlocal mean curvature and symmetry of surfaces.

Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of …

Surveying future location and trajectory prediction methods for moving objects.

Explains a theorem about curves in the plane.

Compactness of metrics with positive sixth order Q-curvature on a sphere with punctures.