arXiv subjects
Publications and source records attributed to Kazuo Akutagawa.
In this paper, we prove the optimal diameter estimate for positive Yamabe metrics on a connected, closed manifold. It is a scalar curvature version of both classical Myers' diameter estimate and Cheng's maximal diameter theorem on Ricci curvature.
An existence result is shown for the asymptotic Dirichlet problem for harmonic maps from the product of the hyperbolic planes to the hyperbolic space, where the Dirichlet data is given on the distinguished boundary (the product of the circles at infinity).
We prove that a minimizer of the Yamabe functional does not exist for a sphere $\mathbb{S}^n$ of dimension $n \geq 3$, endowed with a standard edge-cone spherical metric of cone angle greater than or equal to $4π$, along a great circle of codimension two. When the cone angle along the singularity is smaller than $2π$, the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.
We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$ Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0 $$ where $g_{\mathbb S}$ denotes the standard round metric on $S^4$, then, up to rescaling, $g$ is isometric to $g_{\mathbb S}$. This is an extension of Gursky's gap theorem for positive Einstein metrics on the four-sphere.
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.
The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagrangian surface with finite total curvature in the complex two-space. Moreover, we prove that if the Gauss map of a complete minimal Lagrangian surface which is not a Lagrangian plane omits three values, then it takes all other values infinitely many times.
Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.
We study the regularity properties for solutions of a class of Schrödinger equations $(Δ+ V) u = 0$ on a stratified space $M$ endowed with an iterated edge metric. The focus is on obtaining optimal Hölder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential.
We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there is phrased in terms of a strict inequality of the global Yamabe invariant with a `local Yamabe invariant', which captures information about the local singular structure. All of this is generalized here to the setting of Dirichlet spaces which admit a Sobolev inequality and satisfy a few other mild hypotheses. Applications include a new approach to the nonspherical part of the CR Yamabe problem.
We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik \cite{AB} on the Yamabe problem on spaces with isolated conic singularities.
We consider a complete biharmonic immersed submanifold $M$ in an Euclidean space $\mathbb{E}^N$. Assume that the immersion is proper, that is, the preimage of every compact set in $\mathbb{E}^N$ is also compact in $M$. Then, we prove that $M$ is minimal. It is considered as an affirmative answer to the global version of Chen's conjecture for biharmonic submanifolds.
In this paper, we consider a closed 3-manifold $M$ with flat conformal structure $C$. We will prove that, if the Yamabe constant of $(M, C)$ is positive, then $(M, C)$ is Kleinian.
We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order to prove it, we also solve the orbifold Yamabe problem under a certain condition. We use these results to give some exact computations of the Yamabe invariant of compact orbifolds.
The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schrödinger operator has a finite or infinite number of the discrete pectrum. In this paper, we will give a generalization of this lemma on Euclidean spaces to that on large classes of complete noncompact manifolds. Replacing Euclidean spaces by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some criterions for the above-type question.
In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda-bar. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals + infinity whenever the Yamabe invariant is positive.
We complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\RP^3$, by showing that such manifolds are either $S^3$ or finite connected sums $# m(S^2 \times S^1) # n(S^2 \tilde{\times} S^1)$ for $m + n \geq 1$, where $S^2 \tilde{\times} S^1$ is the nonorientable $S^2$-bundle over $S^1$. A key ingredient is Aubin's Lemma, which says that if the Yamabe constant is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green's functions for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.
For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is strictly less than the Yamabe invariant of $S^{m+n}$ provided $n\geq 2$. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.
We study the Yamabe invariants of cylindrical manifolds and compact orbifolds with a finite number of singularities, by means of conformal geometry and the Atiyah-Patodi-Singer $L^2$-index theory. For an $n$-orbifold $M$ with singularities $Σ_Γ = \{(\check{p}_1, Γ_1), ..., (\check{p}_s, Γ_s)\}$ (where each group $Γ_j<O(n)$ is of finite order), we define and study the \emph{orbifold Yamabe invariant} $Y^{\orb}(M)$. We prove that $Y^{\orb}(M)$ coincides with the corresponding $h$-$\emph{cylindrical Yamabe invariant}$ $Y^{h\textrm{-}\cyl}(M \setminus \{\check{p}_1, ..., \check{p}_s\})$ defined by the authors \cite{AB2}, where $h = h_{Γ_j}$ is the standard metric on the slice $S^{n-1}/Γ_j$ of each end with infinity $\check{p}_j$. Using this, we show that $Y^{\orb}(M)$ is bounded by $Y(S^n) /d$ from above, where $d=\max_j|Γ_j|^{\frac{2}{n}}$. For a cylindrical 4-manifold $X$ with a general slice metric $h$ on the end, we also establish a method for estimating the $h$-cylindrical Yamabe invariant $Y^{h\textrm{-}\cyl}(X)$ from above, in terms of the geometry and topology of $X$. We conclude by an explicit estimate of $Y^{h\textrm{-}\cyl}(X)$ for particular cylindrical 4-manifolds $X$, including that of $Y^{\orb}(M)$ for 4-orbifolds $M$.