arXiv research

Ozawa solution describes surface deformation from Davey-Stewartson II equation.

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) = \max\{tr(K_1),tr(K_2)\}$, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any…

Paper introduces an invariant to distinguish handlebody-knot exteriors.

New invariants measure how far spanning surfaces are from being compressible.

This paper has been withdrawn by the author. The author found that the main results here were already obtained by K. Taniyama and A. Yasuhara `On $C$-distance of knots. Kobe J. Math. 11 (1994), no. 1, 117--127. MR1309997 (95j:57010)'. He would like to thank M. Ozawa for the information.

Brunnian theta curves in 3D spheres have hyperbolic exteriors.

This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the sam…

The JSJ decomposition helps classify genus two handlebody-knots.

New examples of knots with special bridge positions found.

We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is ${\rm trunk}(K) \geq n \cdot {\rm trunk}(J)$ where ${\rm trunk}(\cdot)$ denotes the trunk of a knot, $K$ is a satellite knot with companion $J$, and …

H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a conta…

An automorphism $f$ of a closed orientable surface $Σ$ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, Σ)$ with respect to some embedding $Σ\hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $Σ$ is extendable over $S^3$, then $f$ extends to …

Study on cylindrical handlebody-knots with symmetry and rigidity properties.

In this paper, we investigate the first eigenvalues of two closed eigenvalue problems of the bi-Beltrami-Laplacian on minimal embedded isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although many mathematicians want to derive the corresponding results for the first eigenvalues of bi-Beltrami-Lapla…

Compact Dupin hypersurfaces without constant Lie curvatures found.

The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

Unique ancient solutions found for anisotropic curve shortening flow.

The paper constructs solutions to a critical Dirac equation on spheres.

Paper classifies ancient solutions to 3D Ricci flow.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Study higher-dimensional Ricci flow solutions, proving uniqueness.

Ancient solutions of Ricci flow with Type I growth are classified.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.

New ancient solutions found for curvature flow in 2D.

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient so…

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

Generic level sets in mean curvature flow are BV solutions.

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3-manifolds. We show that any geometric solution is an algebraic one, and we prove the uniqueness of the geometric solutions. Then we do explicit calculations for three inter…

Novikov equation symmetries, solutions, and pseudo-spherical surfaces studied.

Ancient solutions to mean curvature flow have unique shapes.

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. Th…

Necessary and sufficient conditions are provided for a class of warped product manifolds with non-vanishing flux to be supersymmetric solutions of 11D supergravity. Many noncompact, but complete solutions can be obtained in this manner, including the multi-membrane solution initially found by Duff and Stelle. In a diff…

In an ordinary feature selection procedure, a set of important features is obtained by solving an optimization problem such as the Lasso regression problem, and we expect that the obtained features explain the data well. In this study, instead of the single optimal solution, we consider finding a set of diverse yet nea…

Paper finds singular solutions for a specific physics problem on a sphere.

Ancient pancakes solve mean curvature flow problem.

Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…

Study properties of solutions with singularities in the negative cone.

Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…

We consider solutions of Kapustin-Witten equation with Nahm pole boundary on $S^3\times \mathbb{R}^+$. These solutions are usually called Nahm pole solutions. In this paper, we will prove that there exists a constant $C>0$ such that $\|F_A\|_{L^2}\leq C$ for any Nahm pole solution $(A,φ)$.

Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.