共查询到20条相似文献,搜索用时 15 毫秒
1.
Van Hoang Nguyen 《Annals of Global Analysis and Geometry》2018,54(2):237-256
Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)
相似文献
$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$
2.
Let \(\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \(\mathbb {H}^{n}\). We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group \(\mathbb {H}^{n}\). Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space \({ HW}^{1,Q}(\mathbb {H}^{n}) \) on the entire Heisenberg group \(\mathbb {H}^{n}\). Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in \(\mathbb {H}^{n}\) by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space \(\mathbb {H}^n\) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá–Szegö inequality. Due to the absence of the Polyá–Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on \(\mathbb {H}^{n}\): with nonlinear terms f of maximal exponential growth \(\exp (\alpha t^{\frac{Q}{Q-1}})\) as \(t\rightarrow +\infty \). All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.
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$$\begin{aligned} -\mathrm {div}\left( \left| \nabla _{\mathbb {H}}u\right| ^{Q-2} \nabla _{\mathbb {H}}u\right) +V(\xi ) \left| u\right| ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$
3.
A hypermatrix (tensor) complementarity problem \(\textit{HMCP}(q,\mathcal {A})\) is to find a vector \(x\in \mathbb {R}^n\) such that \(x\ge 0,~\mathcal {A}x+q\ge 0,~x^T(\mathcal {A}x+q)=0,\) for every \(q\in \mathbb {R}^n\), where \(\mathcal {A}\) is an mth order hypermatrix (tensor) (Song and Qi in J Optim Theory Appl 165(3): 854–873, 2015). Uniqueness, feasibility, and strict feasibility of the solution of a complementarity problem induced by a (compact) set of hypermatrices are characterized in terms of the hypermatrices involved. 相似文献
4.
Ly Kim Ha 《Annals of Global Analysis and Geometry》2017,51(4):327-346
Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).
相似文献
$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$
5.
On the Stationary Navier–Stokes Problem in $$\mathbb {R}^3$$: An Approach in Weighted Sobolev Spaces
In this paper, we study the steady-state Navier–Stokes equations in \(\mathbb {R}^3\). First, we establish the existence of very weak solution in \(\varvec{L}^p(\mathbb {R}^3)\) with \(3/2< p < +\infty \) under smallness conditions on the data. A uniqueness result is also given in case the data belong to \(\mathbb {L}^r(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) with \(3/2<r<3\). We also discuss the case where the data are not necessarily small. In particular, these results enhance those obtained by Bjorland et al. (Commun Partial Differ Equ 26:216–246, 2011), and are in agreement with those obtained by Kim and Kozono (J Math Anal Appl 395(2):486–495, 2012). Second, we prove a result of existence and uniqueness of weak solution in the weighted Sobolev space \(\varvec{W}_0^{1,p}(\mathbb {R}^3)\cap \varvec{W}_0^{1,\,3/2}(\mathbb {R}^3)\) in case of small external forces given by \(\mathrm{div}\mathbb {F}\) with \(\mathbb {F} \in \mathbb {L}^p(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) and \(1<p<3\). 相似文献
6.
Monica Musso 《Journal of Fixed Point Theory and Applications》2017,19(1):345-361
We consider the equation \(-\Delta u = |u| ^{\frac{4}{n-2}}u + \varepsilon f(x) \) under zero Dirichlet boundary conditions in a bounded domain \(\Omega \) in \(\mathbb {R}^{n}\), \(n \ge 3\), with \(f\ge 0\), \(f\ne 0\). We find sign-changing solutions with large energy. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem recently constructed in del Pino et al. (J Differ Equ 251(9):2568–2597, 2011).
相似文献
$$\begin{aligned} -\Delta w = |w|^{\frac{4}{n-2}} w, \quad w \in {\mathcal D}^{1,2} (\mathbb {R}^n) \end{aligned}$$
7.
Let \(\Omega \subset {\mathbb R}\) be a compact set with measure 1. If there exists a subset \(\Lambda \subset {\mathbb R}\) such that the set of exponential functions \(E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}\) is an orthonormal basis for \(L^2(\Omega )\), then \(\Lambda \) is called a spectrum for the set \(\Omega \). A set \(\Omega \) is said to tile \({\mathbb R}\) if there exists a set \(\mathcal T\) such that \(\Omega + \mathcal T = {\mathbb R}\), the set \(\mathcal T\) is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum. 相似文献
8.
Yoshinori Hashimoto 《Annals of Global Analysis and Geometry》2018,53(2):251-264
Let \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) be a Kähler manifold obtained by blowing up a complex projective space \(\mathbb {P}^n\) along a line \(\mathbb {P}^1\). We prove that \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) is slope unstable with respect to any polarisation, and hence, it does not admit constant scalar curvature Kähler metrics in any rational Kähler class. 相似文献
9.
The Ramanujan sequence \(\{\theta _{n}\}_{n \ge 0}\), defined as \(\theta _{0}= {1}/{2}\), \({n^{n}} \theta _{n}/{n !} = {e^{n}}/{2} - \sum _{k=0}^{n-1} {n^{k}}/{k !}\, \), \(n \ge 1\), has been studied on many occasions and in many different contexts. Adell and Jodrá (Ramanujan J 16:1–5, 2008) and Koumandos (Ramanujan J 30:447–459, 2013) showed, respectively, that the sequences \(\{\theta _{n}\}_{n \ge 0}\) and \(\{4/135 - n \cdot (\theta _{n}- 1/3 )\}_{n \ge 0}\) are completely monotone. In the present paper, we establish that the sequence \(\{(n+1) (\theta _{n}- 1/3 )\}_{n \ge 0}\) is also completely monotone. Furthermore, we prove that the analytic function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} / n^{\alpha }\) is universally starlike for every \(\alpha \ge 1\) in the slit domain \(\mathbb {C}\setminus [1,\infty )\). This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by Ruscheweyh et al. (Israel J Math 171:285–304, 2009), namely that the function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} \) is universally convex. 相似文献
10.
Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic. 相似文献
11.
Kamal Ould Bouh 《Monatshefte für Mathematik》2016,180(4):713-730
In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K. 相似文献
12.
It is often possible to speed up the mixing of a Markov chain \(\{ X_{t} \}_{t \in \mathbb {N}}\) on a state space \(\Omega \) by lifting, that is, running a more efficient Markov chain \(\{ \widehat{X}_{t} \}_{t \in \mathbb {N}}\) on a larger state space \(\hat{\Omega } \supset \Omega \) that projects to \(\{ X_{t} \}_{t \in \mathbb {N}}\) in a certain sense. Chen et al. (Proceedings of the 31st annual ACM symposium on theory of computing. ACM, 1999) prove that for Markov chains on finite state spaces, the mixing time of any lift of a Markov chain is at least the square root of the mixing time of the original chain, up to a factor that depends on the stationary measure of \(\{X_t\}_{t \in \mathbb {N}}\). Unfortunately, this extra factor makes the bound in Chen et al. (1999) very loose for Markov chains on large state spaces and useless for Markov chains on continuous state spaces. In this paper, we develop an extension of the evolving set method that allows us to refine this extra factor and find bounds for Markov chains on continuous state spaces that are analogous to the bounds in Chen et al. (1999). These bounds also allow us to improve on the bounds in Chen et al. (1999) for some chains on finite state spaces. 相似文献
13.
We show that several theorems about Polish spaces, which depend on the axiom of choice (\(\mathcal {AC}\)), have interesting corollaries that are theorems of the theory \(\mathcal {ZF} + \mathcal {DC}\), where \(\mathcal {DC}\) is the axiom of dependent choices. Surprisingly it is natural to use the full \(\mathcal {AC}\) to prove the existence of these proofs; in fact we do not even know the proofs in \(\mathcal {ZF} + \mathcal {DC}\). Let \(\mathcal {AD}\) denote the axiom of determinacy. We show also, in the theory \(\mathcal {ZF} + \mathcal {AD} + V = L(\mathbb {R})\), a theorem which strenghtens and generalizes the theorem of Drinfeld (Funct Anal Appl 18:245–246, 1985) and Margulis (Monatshefte Math 90:233–235, 1980) about the unicity of Lebesgue’s measure. This generalization is false in \(\mathcal {ZFC}\), but assuming the existence of large enough cardinals it is true in the model \(\langle L(\mathbb {R}),\in \rangle \). 相似文献
14.
Christian Ikenmeyer 《Journal of Algebraic Combinatorics》2016,43(1):1-10
We develop structural insights into the Littlewood–Richardson graph, whose number of vertices equals the Littlewood–Richardson coefficient \(c_{\lambda ,\mu }^{\nu }\) for given partitions \(\lambda \), \(\mu \), and \(\nu \). This graph was first introduced in Bürgisser and Ikenmeyer (SIAM J Discrete Math 27(4):1639–1681, 2013), where its connectedness was proved. Our insights are useful for the design of algorithms for computing the Littlewood–Richardson coefficient: We design an algorithm for the exact computation of \(c_{\lambda ,\mu }^{\nu }\) with running time \(\mathcal {O}\big ((c_{\lambda ,\mu }^{\nu })^2 \cdot {\textsf {poly}}(n)\big )\), where \(\lambda \), \(\mu \), and \(\nu \) are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge t\) whose running time is \(\mathcal {O}\big (t^2 \cdot {\textsf {poly}}(n)\big )\). Even the existence of a polynomial-time algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge 2\) is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King et al. (Symmetry in physics. American Mathematical Society, Providence, 2004), stating that \(c_{\lambda ,\mu }^{\nu }=2\) implies \(c_{M\lambda ,M\mu }^{M\nu } = M+1\) for all \(M \in \mathbb {N}\). Here, the stretching of partitions is defined componentwise. 相似文献
15.
Let \(v = (v_1, \ldots , v_n)\) be a vector in \(\mathbb {R}^n {\setminus } \{ 0 \}\). Consider the Laplacian on \(\mathbb {R}^n\) with drift \(\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )\) and the measure \(d\mu (x) = e^{2 \langle v, x \rangle } dx\), with respect to which \(\Delta _{v}\) is self-adjoint. Let d and \(\nabla \) denote the Euclidean distance and the gradient operator on \(\mathbb {R}^n\). Consider the space \((\mathbb {R}^n, d, d\mu )\), which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform \(\nabla (- \Delta _{v} )^{-\frac{1}{2}}\) and for the heat maximal operator, with respect to \(d\mu \). Further, we prove that the uncentered Hardy–Littlewood maximal operator is bounded on \(L^p\) for \(1 < p \le +\infty \) but not of weak type (1, 1) if \(n \ge 2\). 相似文献
16.
Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$
Hidetaka Hamada Mihai Iancu Gabriela Kohr 《Complex Analysis and Operator Theory》2016,10(5):1045-1080
In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators. 相似文献
17.
He-Jun Sun 《Archiv der Mathematik》2018,110(3):291-303
Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \). 相似文献
18.
We study isometric cohomogeneity one actions on the \((n+1)\)-dimensional Minkowski space \(\mathbb {L}^{n+1}\) up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on \(\mathbb {L}^{n+1}\) whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on \(\mathbb {L}^{n+1}\), \(n \ge 3\), which are orbit-equivalent on the complement of an n-dimensional degenerate subspace \(\mathbb {W}^n\) of \(\mathbb {L}^{n+1}\) and not orbit-equivalent on \(\mathbb {W}^n\). We classify isometric cohomogeneity one actions on \(\mathbb {L}^2\) and \(\mathbb {L}^3\) up to orbit-equivalence. 相似文献
19.
Hans-Bert Rademacher 《Journal of Fixed Point Theory and Applications》2017,19(1):289-298
The existence of two geometrically distinct closed geodesics on an n-dimensional sphere \(S^n\) with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [7] and the author [25]. We simplify the proof of this statement by the following observation: If for some \(N \in \mathbb {N}\) all closed geodesics of index \(\le \)N of a non-reversible and bumpy Finsler metric on \(S^n\) are geometrically equivalent to the closed geodesic c, then there is a covering \(c^r\) of minimal index growth, i.e., for all \(m \ge 1\) with \(\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.\) But this leads to a contradiction for \(N =\infty \) as pointed out by Goresky and Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large \(L>0\), we obtain on \(S^2\) a metric of positive flag curvature carrying only two closed geodesics of length \(<L\) which do not intersect.
相似文献
$$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$
20.
We prove that an n-dimensional, \(n\ge 4\), compact gradient shrinking Ricci soliton satisfying a \(L^{\frac{n}{2}}\)-pinching condition is isometric to a quotient of the round \(\mathbb {S}^n\), which improves the rigidity theorem given by Catino (Integral pinched shrinking Ricci solitons, 2016), in dimension \(4\le n\le 6\). 相似文献