前田 瞬

In this paper, we completely classify nontrivial nonflat three dimensional complete shrinking and steady gradient Yamabe solitons without any assumptions. We also give examples of complete expanding gradient Yamabe solitons. Furthermore, we give a proof of the classification of nontrivial two dimensional complete gradient Yamabe solitons without any assumptions.

Abstract 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.

In this paper, we solve the Yamabe soliton version of the Perelman conjecture. We show that any nontrivial complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric.

<title>Abstract</title> We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for <italic>f</italic> -biharmonic Riemannian submersions is also presented.

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

Theory of harmonic maps has been applied into various fields in differential geometry. By extending the notion of harmonic maps, J. Eel Is and L. Lemaire introduced polyharmonic maps of order k. In 1989, S. B. Wang showed the Euler Lagrange equation of polyharmonic maps of order 3 (triharmonic maps). In this paper, we study triharmonic immersion into a sphere. We show the necessary and sufficient condition of triharmonic isometric immersion such that Sigma(m)(s, t=1) del 1/UB(e(s), e(t)) = 0, for all vector field U, and give some non trivial examples. Moreover, we also construct non harmonic triharmonic maps.

<p>We consider polyharmonic maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon left-parenthesis upper M comma g right-parenthesis right-arrow double-struck upper E Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo>:</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi :(M,g)\rightarrow \mathbb {E}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a complete Riemannian manifold into the Euclidean space and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a real constant satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 less-than-or-equal-to p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2\leq p&gt;\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper M Endscripts StartAbsoluteValue upper W Superscript k minus 1 Baseline EndAbsoluteValue Superscript p Baseline d v Subscript g Baseline greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _M|W^{k-1}|^{p}dv_g&gt;\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper M Endscripts StartAbsoluteValue ModifyingAbove nabla With bar upper W Superscript k minus 2 Baseline EndAbsoluteValue squared d v Subscript g Baseline greater-than normal infinity comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mover> <mml:mi mathvariant="normal">∇<!-- ∇ --></mml:mi> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _M|\overline \nabla W^{k-2}|^2dv_g&gt;\infty ,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polyharmonic map of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k minus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i i right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(ii)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper M Endscripts StartAbsoluteValue upper W Superscript k minus 1 Baseline EndAbsoluteValue Superscript p Baseline d v Subscript g Baseline greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _M|W^{k-1}|^{p}dv_g&gt;\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Vol left-parenthesis upper M comma g right-parenthesis equals normal infinity"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Vol</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm {Vol}(M,g)=\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polyharmonic map of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k minus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript s Baseline equals normal upper Delta overbar Superscript s minus 1 Baseline tau left-parenthesis phi right-parenthesis left-parenthesis s equals 1 comma 2 comma midline-horizontal-ellipsis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mover> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W^s=\overline \Delta ^{s-1}\tau (\phi )\ (s=1,2,\cdots )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript 0 Baseline equals phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>ϕ<!-- ϕ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">W^0=\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a corollary, we give an affirmative partial answer to the generalized Chen conjecture.</p>

<title>Abstract</title>We give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian <italic>H</italic>-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.

In [4], J. Eells and L. Lemaire introduced k-energy and k-harmonic maps. In 1989, S.B. Wang [17] showed the first variation formula of the k-energy. In this paper, we give the second variation formula of k-energy and a notion of weakly stable and unstable. We also study k-harmonic maps into product Riemannian manifolds and k-harmonic curves into Riemannian manifolds with constant sectional curvature. Moreover, we give some non-trivial solutions of 3-harmonic curves.

B.Y. Chen introduced biharmonic submanifolds in Euclidean spaces and raised the conjecture "Any biharmonic submanifold is minimal". In this article, we show some affirmative partial answers of generalized Chen's conjecture. Especially, we show that the triharmonic hyper-surfaces with constant mean curvature are minimal.