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Naoya Ando

Bio: Naoya Ando is an academic researcher from Kumamoto University. The author has contributed to research in topics: Curvature & Umbilical point. The author has an hindex of 6, co-authored 21 publications receiving 78 citations. Previous affiliations of Naoya Ando include Max Planck Society & University of Tokyo.

Papers
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Naoya Ando1
TL;DR: In this paper, it was shown that the origin of R3 is an isolated umbilical point of the graph of a homogeneous polynomial in two real variables of degree k ≥ 3.
Abstract: Suppose that the origin o of R3 is an isolated umbilical point of the graph of a homogeneous polynomial in two real variables of degree k≥3. Then we see that the index of o is an element of the set 1−k/2+i[k/2]i=0. Moreover, we see that each element of 1−k/2+i[k/2]i=0 may be the index of o on the graph of a suitable homogeneous polynomial of degree k.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the behavior of principal distributions around the origin of R 3 on condition that the norm of the gradient vector field of log F is bounded from below by a positive constant on a punctured neighborhood of ð0; 0Þ.
Abstract: Let F be a smooth function of two variables which is zero at ð0; 0Þ and positive on a punctured neighborhood of ð0; 0Þ. Then the function expð� 1=F Þ is smoothly extended to ð0; 0Þ and then the origin o of R 3 is an umbilical point of its graph. In this paper, we shall study the behavior of the principal distributions around o on condition that the norm of the gradient vector field of log F is bounded from below by a positive constant on a punctured neighborhood of ð0; 0Þ.

7 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a connected, complete, real-analytic, embedded, parallel curved surface such that any umbilical point is isolated is also homeomorphic to a sphere, a plane, a cylinder or a torus.
Abstract: A surface $S$ in $R^{3}$ is called parallel curved if there exists a plane such that at each point of $S$, there exists a principal direction parallel to this plane. In [2], we studied real-analytic, parallel curved surfaces and in particular, we showed that a connected, complete, real-analytic, embedded, parallel curved surface is homeomorphic to a sphere, a plane, a cylinder, or a torus. In the present paper, we shall show that a connected, complete, embedded, parallel curved surface such that any umbilical point is isolated is also homeomorphic to a sphere, a plane, a cylinder or a torus. However, we shall also show that for each non-negative integer $g\in N\cup\{0\}$, there exists a connected, compact, orientable, embedded, parallel curved surface of genus $g$.

7 citations

Journal ArticleDOI
Naoya Ando1
TL;DR: In this paper, the authors studied the behavior of the principal distributions on a real-analytic, parallel curved surface and classified the connected, complete, real analytic, embedded, and parallel curved surfaces.
Abstract: A smooth surface S in R 3 is called parallel curved if there exists a plane in R 3 such that at each point of S, there exists a principal direction parallel to the plane. For example, a plane, a cylinder and a round sphere are parallel curved. More generally, a surface of revolution is also parallel curved. The purposes of this paper are to study the behavior of the principal distributions on a real-analytic, parallel curved surface and to classify the connected, complete, real-analytic, embedded, parallel curved surfaces.

7 citations


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TL;DR: In this article, a rigidity result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner was generalized to convex C-hypersurfaces.
Abstract: We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex C-hypersurfaces. We apply these results to prove C-convergence of inverse F -curvature flows in the sphere to an equator in S for embedded, closed, orientable, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.

62 citations

Journal ArticleDOI
22 Mar 2021
TL;DR: A list of 33 problems that I have found along the years in my research can be found in this paper, including Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials and some recreational problems with a dynamical component.
Abstract: The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years in my research. I believe that it is worth to think about them and, hopefully, solve some of the problems or make some substantial progress. Many of them are about planar differential equations but there are also questions about other mathematical aspects: Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials and some recreational problems with a dynamical component.

19 citations

Posted Content
TL;DR: In this article, the authors studied a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy, which is the integral of an energy density that depends on the normal at each point over the considered hypersurface.
Abstract: We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hypersurfaces enclosing the same (n+1)-dimensional volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this energy for volume-preserving variations are not smooth in general. We prove that, if the anisotropic energy density function is twice continuously differentiable and convex, then any closed stable equilibrium hypersurface is (up to rescaling) the Wulff shape. We also give fundamental definitions, many examples, and generalizations of well-known concepts and formulas like Steiner's formula and Minkowski's formula to the anisotropic case.

7 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the behavior of principal distributions around the origin of R 3 on condition that the norm of the gradient vector field of log F is bounded from below by a positive constant on a punctured neighborhood of ð0; 0Þ.
Abstract: Let F be a smooth function of two variables which is zero at ð0; 0Þ and positive on a punctured neighborhood of ð0; 0Þ. Then the function expð� 1=F Þ is smoothly extended to ð0; 0Þ and then the origin o of R 3 is an umbilical point of its graph. In this paper, we shall study the behavior of the principal distributions around o on condition that the norm of the gradient vector field of log F is bounded from below by a positive constant on a punctured neighborhood of ð0; 0Þ.

7 citations