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Shun Maeta

Publications and source records attributed to Shun Maeta.

At least 19 recordsLinked to original sources
arXiv2026

Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature

In this paper, we show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal. We also prove that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature on each connected component. This gives partial affirmative answers to Chen's conjecture, to the generalized Chen's conjecture in hyperbolic spaces, and to the Balmu\c{s}-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.

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arXiv2026

\lambda-biharmonic Riemannian submersions from manifolds with constant sectional curvature

In this paper, we study \lambda-biharmonic Riemannian submersions, which generalize biharmonic Riemannian submersions. We prove nonexistence results for \lambda-biharmonic Riemannian submersions from (n + 1)-dimensional Riemannian manifolds with constant sectional curvature c to n-dimensional Riemannian manifolds. Our results show that the critical value \lambda = 2(n - 1)c plays a decisive role. We prove that if c \ge 0, or if c < 0 and \lambda \ne 2(n - 1)c, then any \lambda-biharmonic Riemannian submersion must be harmonic. On the other hand, when \lambda = 2(n - 1)c with c < 0, we construct explicit examples of \lambda-biharmonic Riemannian submersions.

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arXiv2026

Biharmonic rotational surfaces in the four-dimensional Euclidean space are minimal

In this paper, we show that any biharmonic simple rotational surface in the four-dimensional Euclidean space is minimal. The proof is based on reducing the biharmonic equation to a system of ordinary differential equations for the profile curve and then excluding all possible non-minimal branches. This is a partial affirmative answer to Chen's conjecture.

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arXiv2025

Complete gradient Einstein-type Sasakian manifolds with $\alpha=0$

Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons, and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants $\alpha, \beta, \mu$ and $\rho$. In this paper, we show that when $\alpha = 0$, complete gradient Einstein-type Sasakian manifolds are trivial or isometric to the unit sphere. As a consequence, many geometric solitons on Sasakian manifolds turn out to be trivial or isometric to the unit sphere.

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arXiv2025

Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvature

In 2011, Wang and Ou (Math. Z. {\bf 269}:917-925, 2011) showed that any biharmonic Riemannian submersion from a 3-dimensional Riemannian manifold with constant sectional curvature to a surface is harmonic. In this paper, we generalize the 3-dimensional setting to arbitrary dimensions. By constructing an adapted orthonormal frame, we simplify the biharmonic equation for Riemannian submersions and analyze the curvature properties of Riemannian manifolds with constant sectional curvature. As a result, we prove that a Riemannian submersion from an $(n+1)$-dimensional Riemannian manifold with constant sectional curvature to an $n$-dimensional Riemannian manifold is biharmonic if and only if it is harmonic. This result may also be viewed as an affirmative codimension-one Riemannian submersion analogue of Chen's conjecture, the generalized Chen's conjecture, and the BMO conjecture.

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arXiv2025

Complete quasi-Yamabe gradient solitons with bounded scalar curvature

In this paper, we classify complete, nontrivial shrinking and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded below by the soliton constant. We also classify complete, nontrivial expanding and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded above by the soliton constant.

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arXiv2025

Classification of gradient Einstein-type Kähler manifolds with $α=0$

Thanks to the ambitious project initiated by Catino, Mastrolia, Monticelli and Rigoli, which aims to provide a unified viewpoint for various geometric solitons, many classes, including Ricci solitons, Yamabe solitons, $k$-Yamabe solitons, quasi-Yamabe solitons, and conformal solitons, can now be studied under a unified framework known as Einstein-type manifolds. Einstein-type manifolds are characterized by four constants, denoted by $α, β, μ$ and $ρ$. In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with $α= 0$. As a corollary, rotational symmetry for many classes is obtained. In particular, we show that any non-trivial complete gradient quasi-Yamabe soliton on Kähler manifolds is rotationally symmetric.

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arXiv2023

Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

Let $(M,g,f)$ be a 3-dimensional complete steady gradient Ricci soliton. Assume that $M$ is rectifiable, that is, the potential function can be written as $f=f(r)$, where $r$ is a distance function. Then, we prove that $M$ is isometric to (1) a quotient of $\mathbb{R}^3$, or (2) the Bryant soliton. In particular, we show that any 3-dimensional complete rectifiable steady gradient Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. Furthermore, we show that any $3$-dimensional complete rectifiable expanding gradient Ricci soliton with positive Ricci curvature is rotationally symmetric.

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arXiv2023

Rotational symmetry of complete shrinking gradient Yamabe solitons

In this paper, we show that any nontrivial complete shrinking gradient Yamabe soliton whose scalar curvature is bounded below by the soliton constant everywhere and is strictly greater than the constant at some point is rotationally symmetric. This assumption is optimal for higher dimensions. This result resolves the Yamabe-soliton analogue of Perelman's conjecture.

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arXiv2023

Structure of generalized Yamabe solitons and its applications

We consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton version of Perelman's conjecture.

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arXiv2022

Classification of conformal solitons in pseudo-Euclidean spaces

In this paper, we completely classify conformal solitons on pseudo-Riemannian hypersurfaces in pseudo-Euclidean spaces arisen from the position vector field. In particular, the classification of Yamabe solitons on pseudo-Riemannian hypersurfaces in pseudo-Euclidean spaces arisen from the position vector field can be obtained.

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arXiv2021

Classification of generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors

We study complete conformal gradient solitons, a class containing gradient Yamabe solitons and many generalized Yamabe-type structures, including gradient almost Yamabe, gradient k-Yamabe, and gradient h-almost Yamabe solitons, and, after a change of the potential function, gradient Einstein-type manifolds with $\alpha=0$ and $\beta\neq0$ (in particular, quasi-Yamabe solitons). In this paper, we classify complete nontrivial locally conformally flat conformal gradient solitons. This result contributes to an analogue of Perelman's conjecture for Yamabe-type solitons. Moreover, we show that under nonnegative scalar curvature, every nonflat soliton is rotationally symmetric. We also obtain classifications assuming the Cotton or Cao-Chen tensor vanishes.

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arXiv2021

Self-similar solutions to the Hesse flow

We define a Hesse soliton, that is, a self-similar solution to the Hesse flow on Hessian manifolds. On information geometry, the $e$-connection and the $m$-connection are important, which do not coincide with the Levi-Civita one. Therefore, it is interesting to consider a Hessian manifold with a flat connection which does not coincide with the Levi-Civita one. We call it a proper Hessian manifold. In this paper, we show that any compact proper Hesse soliton is expanding and any non-trivial compact gradient Hesse soliton is proper. Furthermore, we show that the dual space of a Hesse-Einstein manifold can be understood as a Hesse soliton.

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arXiv2021

Classification of generalized Yamabe solitons in Euclidean spaces

In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, h-almost Yamabe solitons, gradient k-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.

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