• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    HERMITIAN-EINSTEIN METRICS FOR HIGGS BUNDLES OVER COMPLETE HERMITIAN MANIFOLDS?

    2020-04-27 08:11:08DebinLIU劉德斌

    Debin LIU(劉德斌)

    School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China

    E-mail:liudebin@mail.ustc.edu.cn

    Pan ZHANG(張攀)?

    School of Mathematics,Sun Yat-sen University,Guangzhou 510275,China

    E-mail:zhangpan5@mail.sysu.edu.cn

    Abstract In this paper,we solve the Dirichlet problem for the Hermitian-Einstein equations on Higgs bundles over compact Hermitian manifolds.Then we prove the existence of the Hermitian-Einstein metrics on Higgs bundles over a class of complete Hermitian manifolds.

    Key words Higgs bundles;complete Hermitian manifolds;Hermitian-Einstein metric

    1 Introduction

    Let(M,g)be a Hermitian manifold and the corresponding K?hler form is denoted by ω .A Higgs bundle(E,,θ)over M is a holomorphic vector bundle(E,)coupled with a 1-form θ∈?1,0(M,End(E))satisfying=0 and θ∧θ=0,which is called the Higgs field.Higgs bundle was introduced by Hitchin[10]in his study of self-dual equations on a Riemann surface.It has rich structures and plays an important role in many areas including gauge theory,K?hler and hyper-K?hler geometry,group representations and nonabelian Hodge theory.

    For any given Hermitian metric H on a Higgs bundle E,we de fine its Hitchin-Simpson connection[26]to be

    where DHis the Chern connection with respect to H and,and θ?His the adjoint with respect to H.The curvature of this connection is given by

    where FHis the curvature of the Chern connection DHand?His the(1,0)part of DH.We say H is a Hermitian-Einstein metric,if its curvature FH,θsatis fies the Einstein condition:

    The celebrated Donaldson-Uhlernbeck-Yau theorem states that holomorphic vector bundles over compact K?hler manifolds admit Hermitian-Einstein metrics if they are stable.It was proved by Narasimhan and Seshadri[22]for compact Riemann surface,by Donaldson[7]for algebraic manifolds and by Uhlenbeck-Yau[29]for general compact K?hler manifolds.The inverse problem is that a holomorphic bundle admitting such a metric must be polystable(that is a direct sum of stable bundles with the same slope).And this was proved by Kobayashi[12]and L¨ubke[19]independently.This is the so-called Hitchin-Kobayashi correspondence for holomorphic vector bundles over compact K?hler manifolds.There were many interesting generalized Hitchin-Kobayashi correspondences(see the References[1–6,10,11,13–16,20,21,25,30,32],etc.).In the meanwhile,the Dirichlet problem was solved by Donaldson[8]for Hermitian-Einstein metrics over compact K?hler manifolds with non-empty boundary,and many interesting applications were addressed.Li and Zhang[18]generalized Donaldson’s result to the general Hermitian manifolds and considered a class of vortex equations,which generalize the well-known Hermitian-Einstein equations.As for the non-compact case,Ni and Ren[24]proved that a holomorphic vector bundle over a complete non-compact K?hler manifold with a spectral gap admits a Hermitian-Einstein metric if it admits a metric whose failure to be Hermitian-Einstein is in Lpfor p>1.Ni[23]also showed that the same conclusion holds,for example,if the K?hler manifold satis fies a L2-Sobolev inequality,or if it is non-parabolic(i.e.,admits a positive Green’s function).Later,Zhang[33]studied the existence of Hermitian-Einstein metrics for holomorphic vector bundles over complete Hermitian manifolds.At the same time,he[34]also proved the existence of Hermitian Yang-Mills-Higgs metrics for holomorphic vector bundles on a class of complete K?hler manifolds.

    When it comes to the Higgs bundles,Hitchin[10]and Simpson[26]obtained a Higgs bundle version of Donaldson-Unlenbeck-Yau theorem,i.e.,they proved that a Higgs bundle admits a Hermitian-Einstein metric if and only if it is Higgs polystable.Simpson[26]also considered some non-compact K?hler cases.He introduced the concept of analytic stability for Higgs bundles and proved that the analytic stability implies the existence of Hermitian-Einstein metrics.Recently,Zhang et al.[31]showed the existence of Hermitian-Einstein metrics of analytic stable Higgs bundles over complete non-compact Gauduchon manifolds.

    In this paper,we skip the stability conditions,and assume that the spectrum of a holomorphic Laplace operator has a positive lower bound(see[24]and[33])or the holomorphic Laplace operator satis fies the L2-Sobolev inequality(see[23]),we study the existence of Hermitian-Einstein metrics of Higgs bundles over complete Hermitian manifolds.More precisely,we will prove the following theorem.

    Theorem 1.1Let(M,g)be a complete Hermitian manifold of complex dimension m,and(E,,θ)be a Higgs bundle over M,with an initial Hermitian metric H0.

    1)Assume that the holomorphic operatorhas positive first eigenvalue(M)and thatfor some p ≥ 2 and real number λ.Then there exists a Hermitian-Einstein metric H on E.

    This paper is organized as follows.In Section 2,we give some estimates and preliminaries which will be used in the proof of the theorem.In Section 3,using the Hermitian-Einstein flow,we solve the Dirichlet problem for the Hermitian-Einstein equations over a compact Hermitian manifold.In Section 4,we prove the long-time existence of the Hermitian-Einstein flow over a complete Hermitian manifold.At last,we complete the proof of the main theorem in Section 5.

    2 Preliminary Results

    Let(M,g)be a compact Hermitian manifold,and(E,ˉ?,θ)be a rank r Higgs bundle over M with an initial Hermitian metric H0.Denote by ω the K?hler form,and de fine the operator Λ as the contraction with ω,i.e.,for α ∈ ?1,1(M,E),one has Λα =hα,ωi.For any local complex coordinatein M,we can de fine the holomorphic Laplace operator for functions as

    where V is a well-de fined vector fields on M.The holomorphic Laplace operatorcoincides with the usual Beltrami-Laplace operator if and only if the base manifold(M,g)is K?hler.

    Given any Hermitian metric H on E,since the bundle E is holomorphic with the holomorphic structure,there is a unique corresponding connection AHwhich is called the Chern connection.Under a holomorphic local frame{eα},we can express the metric H as a positive de fined matrix()1≤α,β≤r,where=H(eα,eβ).For simplicity we will still denote it by H.Then locally the Chern connection AHand its curvature form FHcan be written as

    Now consider two Hermitian metrics H and K on E.Set h=K?1H ∈?0(M,End(E)).It is easy to check that h is positive de fined and self-adjoint with respect to both H and K.Then by a direct calculation we have

    Next,we turn to a family of Hermitian metrics H(t)on E with an initial metric H(0)=H0.We will follow the classical heat flow method to deduce the existence of Hermitian-Einstein metric.Actually,we consider the following Hermitian-Einstein flow

    Taking a local holomorphic frame{eα}of E and local complex coordinates{zi}of M,the above flow can be written as

    We will see later the following proposition plays an important role in our discussion.

    Proposition 2.1Let H(t)be a solution of the flow(2.1),then

    ProofFor simplicity,setThen by a direct calculation,we have

    and

    Hence

    Now we recall the Donaldson’s distance on the space of Hermitian metrics as follows.

    De finition 2.2For any two Hermitian metrics H and K on the bundle E,we de fine

    where r=rank(E).

    If we choose a local frame to diagonalize H?1K to be diag(λ1,···,λr),then

    from which we can see that σ≥0 with equality if and only if H=K.Let d be the Riemannian distance function on the metric space,then

    holds for some monotone functions f1and f2.So we can conclude from this inequality that a sequence of metrics Hiconverges to some H in C0-topology if and only if

    For later use,we need the following lemma.

    Lemma 2.3

    ProofIt is easy to check that

    On the other hand,it is easy to check that tris nonnegative,by doing calculation locally[27].Hence we have the following proposition.

    Proposition 2.4Let H,K be two Hermitian-Einstein metrics,then

    Next,instead of considering H,K as Hermitian-Einstein metrics,we assume H=H(t),K=K(t)to be two solutions of the Hermitian-Einstein flow(2.1)with the same initial value H0.Similar to Proposition 2.4,we prove the following proposition.

    Proposition 2.5

    ProofSet h(t)=K(t)?1H(t).Notice that

    These two identities together with Lemma 2.3 show that

    For further discussion,we prove the following.

    Proposition 2.6Let H(x,t)be a solution of the Hermitian-Einstein flow with the initial metric H0and set h=H,then

    ProofThe proof is quite straightforward.First notice that

    And following the same argument as in[27],one can show that

    Set d=trh+trh?1,then from the above two inequalities,we can get

    Using the above discussion and Lemma 2.3,we can deduce that

    where the last inequality follows from the fact that

    Proceeding by a similar argument,we have

    Corollary 2.7Let H be a Hermitian-Einstein metric and H0be the initial Hermitian metric.Let h=H,then

    3 Over Compact Manifolds

    In this section our primary purpose is to solve the Dirichlet problem for the Hermitian-Einstein flow over a compact manifold.Speci fi cally,when the base manifold M is closed,we consider the following problem

    And when M is a compact manifold with a non-empty smooth boundary?M,for any given initial metric ? over ?M we instead consider the following boundary value problem

    One can easily check that the equation in(3.1)is non-linear,strictly parabolic.So we get the short-time existence from the standard parabolic PDE theory[9].

    Theorem 3.1For sufficiently small ε>0,problems(3.1)and(3.2)have a smooth solution de fined for 0≤ t< ε.

    Next,following a standard argument,we can show the long-time existence of(3.1)and(3.2).

    Lemma 3.2Suppose that a smooth solution Htto(3.1)or(3.2)is de fined for 0≤t

    Then Htconverges in C0topology to some continuous non-degenerate metric HTas t→T.

    ProofIn order to prove the convergence,it suffices to show that,given any ε>0,we can find δ>0 such that

    And this can be easily seen from the continuity at t=0 combining with Proposition 2.5 and the maximum principle.

    So,it remains to show HTis non-degenerate.By Proposition 2.1,we know that

    where C=C(H0)is a uniform constant.By a direct calculation we have

    And similarly

    Then we can conclude that σ(H,H0)is uniformly bounded on M × [0,T),which implies that HTis non-degenerate.

    For further consideration,we need the following lemma.

    Lemma 3.3Suppose M is a closed Hermitian manifold without boundary(compact with non-empty boundary).Let H(t)for 0≤t

    1)H(t)converges in C0topology to some continuous metric HTas t→T,

    2)supM|ΛFH|H0is uniformly bounded for t

    Then H(t)is bounded in C1,and also bounded in(for any 1

    Since the proof is exactly the same as that in[7]and[26],we omit it here.

    Theorem 3.4Problems(3.1)and(3.2)have a unique solution H(t)which exists for 0≤t<∞.

    ProofTheorem 3.1 guarantees that a solution exists for a short time.Then we suppose that there is a solution H(t)existing for 0≤t

    Finally,since we have proved the long-time existence of the Dirichlet problem(3.2),it remains for us to show that the solution H(t)converges to a metric H∞as the time t approaches to the in fi nity,and that the limit H∞is Hermitian-Einstein.

    Suppose H(t)is a solution to(3.2)for 0≤t<∞.As in the previous section we still setFrom Proposition 2.1 and the fact thatholds for any section α of End(E),we have

    Next,according to Proposition 1.8 of Chapter 5 in[28],the following Dirichlet problem is solvable

    Therefore the maximum principle implies that

    for any y∈M and 0≤t<∞.

    Let 0≤t1≤t<∞,=H?1(x,t1)H(x,t).Obviouslyˉh satis fies

    Then we have

    Integrating it over[t1,t]gives

    From(3.6)and(3.7),we have that H(t)converges in the C0topology to some continuous metric H∞as t?→ +∞.Hence using Lemma 3.3 again we know that H(t)has uniform C1andbounds.This together with the fact thatis uniformly bounded and the standard elliptic regularity arguments shows that,by passing to a subsequence if necessary,H(t)→H∞in C∞topology.And from(3.6)we have

    i.e.,H∞is the desired Hermitian-Einstein metric satisfying the Dirichlet boundary condition.Since for any given initial Hermitian metric ? over boundary ?M one can construct a Hermitian metric H0over M with H0|?M= ? by choosing a proper partition of unity,we have eventually proved the following theorem.

    Theorem 3.5Suppose(M,g)is a compact Hermitian manifold with non-empty boundary?M and(E,,θ)is a Higgs bundle over M.Then for any Hermitian metric ? on restriction of E to?M there is a unique Hermitian metric H on E such that

    4 Hermitian-Einstein Flow Over Complete Hermitian Manifolds

    Let M be a complete non-compact Hermitian manifold without boundary,in which case we will still call it complete for short,and E be a rank r Higgs bundle over M with an initial Hermitian metric H0.Suppose the curvature of AH0satis fiesIdE|H0≤C0for some positive constant C0.What we want to prove is the existence of the long time solution to the Hermitian-Einstein flow over M starting at H0.

    It is well known that for any topological manifold M we can always find a compact exhaustion sequence,i.e.,a countable collection of compact subsetsof M which satis fies?i? ?i+1and?i=M.Moreover,if the manifold is smooth we can further assume that??iis smooth for each i.Now for such an exhaustion,by our discussion in Section 3 we can find Hermitian metrics H(x,t)on E|?ifor each i,which solve the following problems

    Notice here we use H0as both the initial metric with respect to t and the boundary metric.So Proposition 2.1 together with the maximum principle shows that

    As before we set hi=Hi.Then through a simple calculation(as is shown in the proof of Lemma 3.2),we can see that

    Integrating along time direction from 0 to T,we have

    This immediately implies

    and

    In particular,for any compact subset K we can choose i large enough so that K??i,then we get the following C0-estimate

    Without loss of generality,we can assume that K=Bo(R),here K=Bo(R)denotes the geodesic ball of radius R center at a fixed point o∈M.We want to show that,by passing to a subsequence if necessary,{Hi}converges uniformly to a Hermitian metric H∞(x,t)over Bo(R)×[0,T/2].By Lemma 2.3,we have

    Hence using(4.3)and(4.4),we can get the following estimate

    where C3is a uniform constant depending only on C0,T,R and V.Since(4.7)gives the uniform C0-estimate for hiand notice thatis also uniformly bounded,(4.8)implies that hiis uniformly bounded in(Bo(2R)×[0,T]).From Sobolev embedding theorem we know that(Bo(2R)×[0,T])can be compactly embedded into L2(Bo(2R)×[0,T]).So by passing to a subsequence if necessary,we conclude that hi,and hence Hi,converges in L2(Bo(2R)×[0,T]).

    This implies that given any ε>0,we have

    for j,k sufficiently large.In order to show the uniform convergence,we need the following mean-value type inequality.

    Lemma 4.1(see[33,Lemma 5.2])Let M be an m-dimensional complete non-compact Hermitian manifold without boundary,and Bo(2R)be a geodesic ball,centered at o∈M of radius 2R.Suppose that f(x,t)is a non-negative function satisfying

    over Bo(2R)×[0,T].Let?K ≤0 be the lower bound of the Ricci curvature of Bo(2R).Then for any p>0,there exist positive constant C′and C′′depending only on C,m,R,K,p,T and V such that

    Now let f(x,t)=σ2(Hj(x,t),Hk(x,t)).Obviously fsatis fies(4.10)with C=0.Then by Lemma 4.1,(4.9)and the fact that Hj=Hkat t=0 gives

    here C4is a positive constant depending only on C0,R,T and the bound of the sectional curvature on Bo(2R).And this implies that,by passing to a subsequence if necessary,Hiconverges uniformly to a continuous Hermitian metric H∞over Bo(R)×[0,T/2].

    Finally,we want to use the above C0-estimate to deduce a C1-estimate over any compact subset K.We will follow Donaldson[8]and Zhang[33].For any point x∈Bo(2R)we can choose a coordinate ball Bx(R′)centered at p small enough such that E can be locally trivialized over it.Let{yj}be a real coordinate on Bx(R′)and denote?0(M,End(E)).Then we have the following.

    Proposition 4.2ρlis de fined as above,then

    ProofFrom(4.1),we know Hisatis fies

    For now for the sake of calculation convenience,we omit the subscript i and l and denote Hi,ρlto be H,ρ respectively.Then by considering a one-parameter family of solutions obtained by translating in the direction ofone can check that

    where

    As is shown in the proof of Proposition 2.1,one can check that

    By direct calculation,we have

    and one can easily check that

    Then combining those calculations together shows that

    Then this proposition together with Lemma 4.1 and(4.8)gives

    where C5is a positive constant independently on i.Moreover,for the Hermitian metric g there exist constants C6and C7such that

    holds over Bx(R′).Hence we have

    From this and(4.12)we can fi nally conclude that there exists a positive constant C8which is independently of i such that

    Since x is arbitrary,we can see that Hihas a uniform C1-bound over Bo(R)×[0,T/4].Since we have derive the C0and C1estimates,the standard parabolic theory is enough to show the global convergence of Hi,which is

    Theorem 4.3Let(M,g)be a complete non-compact Hermitian manifold without boundary,and(E,,θ)be a Higgs bundle over M with an initial Hermitian metric H0.If the initial data satisfiesfor some positive constant C0,then the following Hermitian-Einstein flow

    has a solution which is de fined on M ×[0,∞).

    5 Hermitian-Einstein Metrics on Complete Hermitian Manifolds

    In this section we will prove the existence of Hermitian-Einstein metric on complete noncompact Hermitian manifold.We will proceed by the direct elliptic method.The argument is similar to that in[33]for holomorphic vector bundle case.The following are the main assumptions we will need.

    De finition 5.1(Positive spectrum) Let M be a complete Hermitian manifold.We say the holomorphic Laplace operatorhas positive first eigenvalue,if there exists a positive number c such that for any compactly supported smooth function φ one has

    De finition 5.2(L2-Sobolev inequality) Let M be a complete Hermitian manifold of complex dimension m.We say the holomorphic Laplace operatorsatis fies L2-Sobolev inequality,if there exists a positive constant SMsuch that for any compact supported smooth function φ one has

    In order to use the direct elliptic method,we now introduce a new distance function instead of σ.For two metrics H and K,we de fine

    The relationship between σ and τ is given by

    Now choose a compact exhaustion sequenceas in Section 4.By our previous discussion in Section 3,we know that over each ?ithere exists a Hermitian metric Hisuch that

    where H0is the given initial metric on E.Denote hi=Hiand τi= τ(H0,Hi).Then by Corollary 2.7 we can see that

    Next we impose the following condition on the holomorphic Laplace operator.

    Condition 5.3There exists a positive number p>0 such that for every non-negative function f∈Lp(M),there exists a non-negative solution u∈C0(M)of

    Theorem 5.4Let(M,g)be a complete Hermitian manifold,and(E,,θ)be a Higgs bundle over M with an initial Hermitian metric H0.Assume for the holomorphic Laplace operator?,Condition 5.3 is satis fied with some positive number p.Assume further thatThen there exists a Hermitian-Einstein metric H on E.

    ProofLet u be a solution to.If M satis fies Condition 5.3,this together with(5.1)and the maximun principle shows that

    holds for any x∈?i.Using this C0-estimate,and proceeding as how we did in Section 4 shows that Hiconverges uniformly over any compact subset of M to a smooth Hermitian metric H satisfying

    over any compact subset.Hence H is a Hermitian-Einstein metric over the whole manifold M.

    Therefore we complete the proof of Theorem 1.1 by the following lemma.

    Lemma 5.5(see[33,Lemma 6.4,Lemma 6.5])Let M be a complete Hermitian manifold of complex dimension m.

    1)Assume the holomorphic Laplace operator??has positive first eigenvalue(M).Then for a non-negative function f∈Lp(M),where p≥2,the following equation

    has a non-negative solution

    2)Assume??satis fies the L2-Sobolev inequality.Then for a non-negative function f∈Lp(M),where 2≤p

    在线播放无遮挡| 精品一区二区三卡| 成人亚洲精品av一区二区| 国产中年淑女户外野战色| 久久国内精品自在自线图片| 蜜桃亚洲精品一区二区三区| 特大巨黑吊av在线直播| 国产成人午夜福利电影在线观看| 久久久精品免费免费高清| 天堂中文最新版在线下载 | 欧美性感艳星| 麻豆乱淫一区二区| 国产 一区 欧美 日韩| 国产精品一区二区三区四区免费观看| 亚洲欧美一区二区三区黑人 | 一级毛片黄色毛片免费观看视频| 国产伦精品一区二区三区视频9| 成人漫画全彩无遮挡| 人妻少妇偷人精品九色| 99久久精品一区二区三区| 高清视频免费观看一区二区| 国产真实伦视频高清在线观看| a级毛片免费高清观看在线播放| 2021少妇久久久久久久久久久| 高清毛片免费看| 不卡视频在线观看欧美| 国产一区二区三区av在线| 国产视频首页在线观看| 久久99热6这里只有精品| 欧美高清性xxxxhd video| 日韩成人伦理影院| 亚洲欧美精品专区久久| 国产高清有码在线观看视频| 国模一区二区三区四区视频| 日本一本二区三区精品| 亚洲av欧美aⅴ国产| 亚洲自偷自拍三级| 中文乱码字字幕精品一区二区三区| 国产高清三级在线| 在线观看av片永久免费下载| 欧美性猛交╳xxx乱大交人| 少妇人妻精品综合一区二区| 人妻一区二区av| 日日啪夜夜撸| 嫩草影院新地址| 欧美丝袜亚洲另类| 日日摸夜夜添夜夜添av毛片| 日产精品乱码卡一卡2卡三| 激情五月婷婷亚洲| 成人二区视频| 天天一区二区日本电影三级| 美女主播在线视频| 国精品久久久久久国模美| 免费看日本二区| 内射极品少妇av片p| 国产精品成人在线| 国产精品爽爽va在线观看网站| 熟女人妻精品中文字幕| 80岁老熟妇乱子伦牲交| 亚洲自拍偷在线| 成人漫画全彩无遮挡| 成年免费大片在线观看| 特级一级黄色大片| 一区二区av电影网| 男插女下体视频免费在线播放| 国产日韩欧美亚洲二区| 亚洲精品aⅴ在线观看| 国产男人的电影天堂91| 一级黄片播放器| 秋霞伦理黄片| 婷婷色麻豆天堂久久| 搡老乐熟女国产| 免费在线观看成人毛片| 大香蕉久久网| 少妇人妻久久综合中文| 国产亚洲精品久久久com| 欧美一级a爱片免费观看看| 久久久久久久久久人人人人人人| 1000部很黄的大片| 女人久久www免费人成看片| 国产又色又爽无遮挡免| 亚洲,欧美,日韩| 一个人看的www免费观看视频| 欧美区成人在线视频| 免费观看在线日韩| 午夜福利高清视频| 久久国产乱子免费精品| 亚洲自偷自拍三级| 国产av国产精品国产| 国产免费一区二区三区四区乱码| 免费黄色在线免费观看| 国产精品久久久久久久久免| 香蕉精品网在线| 免费黄频网站在线观看国产| 尾随美女入室| 三级经典国产精品| 免费观看av网站的网址| 亚洲自拍偷在线| 日韩伦理黄色片| 日韩三级伦理在线观看| 一级毛片 在线播放| 国产男女内射视频| 最近中文字幕高清免费大全6| 一级毛片我不卡| 丝袜脚勾引网站| 蜜桃亚洲精品一区二区三区| 精品久久久噜噜| 国产色爽女视频免费观看| 免费黄网站久久成人精品| 久久久久性生活片| 国产中年淑女户外野战色| 国产在视频线精品| 亚洲精品影视一区二区三区av| 久久97久久精品| 最后的刺客免费高清国语| 免费大片18禁| 亚洲色图综合在线观看| 国产亚洲av嫩草精品影院| 国产视频内射| 成人二区视频| 老女人水多毛片| 人妻 亚洲 视频| 人人妻人人澡人人爽人人夜夜| 免费高清在线观看视频在线观看| 久久久久久伊人网av| 久久久久久久精品精品| 亚洲国产精品成人久久小说| 天堂俺去俺来也www色官网| 国产日韩欧美在线精品| 波野结衣二区三区在线| 人妻制服诱惑在线中文字幕| 永久网站在线| 亚洲国产精品专区欧美| 成人特级av手机在线观看| 欧美xxxx黑人xx丫x性爽| 麻豆乱淫一区二区| 欧美一级a爱片免费观看看| www.av在线官网国产| 精品人妻偷拍中文字幕| 欧美bdsm另类| 久久久精品欧美日韩精品| 一本久久精品| 99热这里只有精品一区| 亚洲精华国产精华液的使用体验| 熟女电影av网| 少妇丰满av| 国产精品三级大全| 婷婷色综合www| 亚洲精品国产av蜜桃| 日韩在线高清观看一区二区三区| 欧美日韩一区二区视频在线观看视频在线 | 成年人午夜在线观看视频| av线在线观看网站| 亚洲,欧美,日韩| 久久精品人妻少妇| 又粗又硬又长又爽又黄的视频| 97热精品久久久久久| 婷婷色av中文字幕| 波多野结衣巨乳人妻| 99视频精品全部免费 在线| 黄片无遮挡物在线观看| 欧美另类一区| 麻豆成人午夜福利视频| 国产色爽女视频免费观看| av福利片在线观看| 久久99精品国语久久久| 国产黄a三级三级三级人| 18禁裸乳无遮挡免费网站照片| 亚洲av不卡在线观看| 色5月婷婷丁香| 午夜日本视频在线| 亚洲内射少妇av| 亚洲欧美清纯卡通| 国产日韩欧美亚洲二区| 女人久久www免费人成看片| 国产一区二区三区av在线| 亚洲精品国产色婷婷电影| 好男人在线观看高清免费视频| 久久精品国产鲁丝片午夜精品| 直男gayav资源| 亚洲,欧美,日韩| 伦理电影大哥的女人| 国产爱豆传媒在线观看| 亚洲av免费在线观看| 最近手机中文字幕大全| 欧美成人午夜免费资源| 少妇高潮的动态图| 亚洲精品成人久久久久久| 男女国产视频网站| 乱系列少妇在线播放| 国产探花在线观看一区二区| 欧美三级亚洲精品| av免费在线看不卡| 能在线免费看毛片的网站| 香蕉精品网在线| 搞女人的毛片| av播播在线观看一区| av在线天堂中文字幕| 免费黄色在线免费观看| 久热这里只有精品99| 在线天堂最新版资源| 少妇人妻久久综合中文| 18禁在线无遮挡免费观看视频| 熟妇人妻不卡中文字幕| 免费大片黄手机在线观看| 亚洲成人精品中文字幕电影| 亚洲国产欧美人成| 亚洲国产色片| 久久久久久久亚洲中文字幕| 亚洲电影在线观看av| 97超视频在线观看视频| 亚洲高清免费不卡视频| 中文字幕制服av| 日日摸夜夜添夜夜添av毛片| 亚洲激情五月婷婷啪啪| 欧美少妇被猛烈插入视频| 禁无遮挡网站| 国产日韩欧美亚洲二区| 黄色日韩在线| 国产精品av视频在线免费观看| 亚洲色图综合在线观看| 在线亚洲精品国产二区图片欧美 | 日本免费在线观看一区| 日韩大片免费观看网站| 插阴视频在线观看视频| 真实男女啪啪啪动态图| 伊人久久精品亚洲午夜| 国产乱人视频| 亚洲一级一片aⅴ在线观看| 亚洲欧洲日产国产| 日韩av在线免费看完整版不卡| 国产成人精品婷婷| 一区二区三区乱码不卡18| 男人狂女人下面高潮的视频| 国产伦理片在线播放av一区| 日韩av在线免费看完整版不卡| 日韩av免费高清视频| videossex国产| 蜜桃亚洲精品一区二区三区| 国产伦理片在线播放av一区| 国产黄色免费在线视频| 综合色av麻豆| 99久久精品国产国产毛片| 精品人妻一区二区三区麻豆| 九草在线视频观看| 一区二区三区精品91| 成人国产麻豆网| 一二三四中文在线观看免费高清| 亚洲人与动物交配视频| 久久久a久久爽久久v久久| 午夜福利在线观看免费完整高清在| 国产亚洲91精品色在线| 人妻少妇偷人精品九色| 成年女人在线观看亚洲视频 | 国产成人91sexporn| 色吧在线观看| 好男人视频免费观看在线| 男女那种视频在线观看| 免费观看在线日韩| 欧美精品国产亚洲| 国精品久久久久久国模美| 男人狂女人下面高潮的视频| 黄色配什么色好看| 久久久久久九九精品二区国产| 一本色道久久久久久精品综合| 国语对白做爰xxxⅹ性视频网站| 免费看av在线观看网站| 看免费成人av毛片| 亚洲欧洲国产日韩| 国模一区二区三区四区视频| 国产精品99久久久久久久久| 亚洲国产欧美在线一区| 老师上课跳d突然被开到最大视频| 哪个播放器可以免费观看大片| 国产精品99久久99久久久不卡 | 日韩免费高清中文字幕av| 久久久精品免费免费高清| 精品一区二区免费观看| 日本黄色片子视频| 亚洲真实伦在线观看| 精品一区二区三区视频在线| 国产老妇女一区| 日本爱情动作片www.在线观看| 欧美变态另类bdsm刘玥| 夫妻午夜视频| 新久久久久国产一级毛片| 久久国产乱子免费精品| 成人鲁丝片一二三区免费| 中文天堂在线官网| 国产老妇伦熟女老妇高清| 人体艺术视频欧美日本| 日韩欧美精品免费久久| 女人久久www免费人成看片| 欧美性感艳星| 欧美日本视频| 精品久久国产蜜桃| 秋霞伦理黄片| 麻豆国产97在线/欧美| 丝袜喷水一区| 我的女老师完整版在线观看| 涩涩av久久男人的天堂| 美女国产视频在线观看| 九九爱精品视频在线观看| 中文欧美无线码| 少妇 在线观看| 国产免费福利视频在线观看| 国产亚洲av嫩草精品影院| 黄色一级大片看看| av在线播放精品| 久久亚洲国产成人精品v| 高清av免费在线| videos熟女内射| av免费观看日本| 韩国av在线不卡| 身体一侧抽搐| 男人爽女人下面视频在线观看| 熟女电影av网| 免费观看的影片在线观看| 成人亚洲精品一区在线观看 | 亚洲精品国产av成人精品| 成年版毛片免费区| 欧美成人a在线观看| 九草在线视频观看| 亚洲国产精品成人综合色| 91狼人影院| 噜噜噜噜噜久久久久久91| 日韩电影二区| 国产一区二区亚洲精品在线观看| 亚洲丝袜综合中文字幕| 精品久久国产蜜桃| 国产午夜精品一二区理论片| 91久久精品国产一区二区三区| 可以在线观看毛片的网站| 欧美精品国产亚洲| 美女被艹到高潮喷水动态| 亚洲自偷自拍三级| 男女无遮挡免费网站观看| xxx大片免费视频| av在线老鸭窝| 国产男人的电影天堂91| 成年女人在线观看亚洲视频 | 国产精品99久久久久久久久| 日本爱情动作片www.在线观看| 91狼人影院| 一级二级三级毛片免费看| 97超碰精品成人国产| 精品一区在线观看国产| 欧美日本视频| 午夜免费鲁丝| 国产精品一区二区三区四区免费观看| 91精品伊人久久大香线蕉| 白带黄色成豆腐渣| 天天躁日日操中文字幕| 成人亚洲欧美一区二区av| 色视频在线一区二区三区| 免费人成在线观看视频色| 久久精品久久久久久噜噜老黄| 久久ye,这里只有精品| 一级毛片久久久久久久久女| av在线老鸭窝| 亚洲怡红院男人天堂| 干丝袜人妻中文字幕| 国产爱豆传媒在线观看| 高清欧美精品videossex| 伦理电影大哥的女人| 18禁在线播放成人免费| 直男gayav资源| 中文欧美无线码| 97精品久久久久久久久久精品| 成人免费观看视频高清| 精品一区在线观看国产| 亚洲精品中文字幕在线视频 | 菩萨蛮人人尽说江南好唐韦庄| 日本一二三区视频观看| 国产精品国产av在线观看| 日日啪夜夜撸| 亚洲婷婷狠狠爱综合网| 精品人妻一区二区三区麻豆| 22中文网久久字幕| 97超视频在线观看视频| 欧美3d第一页| 国产黄片美女视频| 国产真实伦视频高清在线观看| 免费观看的影片在线观看| 99视频精品全部免费 在线| 一本久久精品| 国产免费福利视频在线观看| 亚洲人与动物交配视频| 街头女战士在线观看网站| 日本黄色片子视频| 看非洲黑人一级黄片| 熟妇人妻不卡中文字幕| 最近最新中文字幕免费大全7| 国产大屁股一区二区在线视频| 各种免费的搞黄视频| 国产一区有黄有色的免费视频| 国产欧美另类精品又又久久亚洲欧美| 可以在线观看毛片的网站| 一级毛片电影观看| 欧美+日韩+精品| 日韩成人av中文字幕在线观看| 国产一区二区三区综合在线观看 | 99热这里只有精品一区| 九九久久精品国产亚洲av麻豆| av在线app专区| av免费在线看不卡| 国产欧美日韩精品一区二区| 搡老乐熟女国产| 亚洲无线观看免费| 久久韩国三级中文字幕| 少妇高潮的动态图| 久久久精品欧美日韩精品| 国产欧美日韩一区二区三区在线 | 亚洲国产最新在线播放| 中文字幕人妻熟人妻熟丝袜美| 22中文网久久字幕| 日韩成人av中文字幕在线观看| 日本一本二区三区精品| av播播在线观看一区| 六月丁香七月| 日韩欧美一区视频在线观看 | 美女被艹到高潮喷水动态| 欧美日韩精品成人综合77777| 日韩亚洲欧美综合| 国产探花极品一区二区| 亚洲自偷自拍三级| 成人午夜精彩视频在线观看| 少妇裸体淫交视频免费看高清| 国产欧美亚洲国产| 最近手机中文字幕大全| 日韩av在线免费看完整版不卡| 久久精品久久久久久久性| av国产免费在线观看| 亚洲精品一区蜜桃| 日韩av免费高清视频| 亚洲精品一二三| 91在线精品国自产拍蜜月| 人人妻人人澡人人爽人人夜夜| 美女cb高潮喷水在线观看| av网站免费在线观看视频| 国产中年淑女户外野战色| 亚洲熟女精品中文字幕| 中文字幕人妻熟人妻熟丝袜美| 啦啦啦在线观看免费高清www| 80岁老熟妇乱子伦牲交| 免费观看性生交大片5| av福利片在线观看| 人妻少妇偷人精品九色| 成人黄色视频免费在线看| 亚洲精品一区蜜桃| 麻豆国产97在线/欧美| 国产毛片在线视频| 欧美xxxx黑人xx丫x性爽| 免费观看av网站的网址| 少妇熟女欧美另类| 看非洲黑人一级黄片| 中文在线观看免费www的网站| 欧美精品国产亚洲| 人人妻人人爽人人添夜夜欢视频 | 久久久精品94久久精品| 一级黄片播放器| 亚洲成人中文字幕在线播放| 乱码一卡2卡4卡精品| 亚洲欧洲日产国产| 亚洲国产日韩一区二区| 男插女下体视频免费在线播放| 在线观看av片永久免费下载| 女的被弄到高潮叫床怎么办| 国产成年人精品一区二区| 人人妻人人看人人澡| 校园人妻丝袜中文字幕| 国产老妇女一区| 国产精品熟女久久久久浪| av网站免费在线观看视频| 成人二区视频| 亚洲一级一片aⅴ在线观看| 人妻 亚洲 视频| 少妇的逼好多水| 成人综合一区亚洲| 麻豆久久精品国产亚洲av| 欧美日韩视频精品一区| 少妇人妻精品综合一区二区| 在线天堂最新版资源| 黄色怎么调成土黄色| 亚洲国产精品999| 成人特级av手机在线观看| 新久久久久国产一级毛片| 国产极品天堂在线| 男人爽女人下面视频在线观看| 99精国产麻豆久久婷婷| 啦啦啦中文免费视频观看日本| 久久久欧美国产精品| av又黄又爽大尺度在线免费看| 日本三级黄在线观看| 美女主播在线视频| 久久99热6这里只有精品| 婷婷色麻豆天堂久久| 国产精品一区www在线观看| 免费观看av网站的网址| 国模一区二区三区四区视频| 女人久久www免费人成看片| 深爱激情五月婷婷| 人人妻人人爽人人添夜夜欢视频 | 免费看av在线观看网站| 久久久久久伊人网av| 波多野结衣巨乳人妻| 亚洲aⅴ乱码一区二区在线播放| 国产av码专区亚洲av| 一级黄片播放器| 最新中文字幕久久久久| 男女无遮挡免费网站观看| 精品少妇久久久久久888优播| 欧美区成人在线视频| 亚洲精品第二区| 国内揄拍国产精品人妻在线| 午夜精品一区二区三区免费看| 欧美性猛交╳xxx乱大交人| 亚洲经典国产精华液单| 水蜜桃什么品种好| 亚洲av成人精品一二三区| 日韩欧美一区视频在线观看 | 国产精品精品国产色婷婷| 亚洲av二区三区四区| 久久国内精品自在自线图片| 国产精品福利在线免费观看| videos熟女内射| 狂野欧美激情性xxxx在线观看| 欧美激情久久久久久爽电影| 亚洲精品日本国产第一区| 亚洲精品成人久久久久久| 久久久久久久亚洲中文字幕| 黄色怎么调成土黄色| 亚洲电影在线观看av| 啦啦啦啦在线视频资源| 一本—道久久a久久精品蜜桃钙片 精品乱码久久久久久99久播 | 九九久久精品国产亚洲av麻豆| 可以在线观看毛片的网站| av在线播放精品| 国产女主播在线喷水免费视频网站| 欧美 日韩 精品 国产| 亚洲自偷自拍三级| 伊人久久精品亚洲午夜| 国产亚洲av嫩草精品影院| 国产真实伦视频高清在线观看| 亚洲av成人精品一区久久| 中国国产av一级| 在线天堂最新版资源| 观看免费一级毛片| 久久人人爽av亚洲精品天堂 | 国产男人的电影天堂91| 国产欧美亚洲国产| av女优亚洲男人天堂| 视频区图区小说| 免费少妇av软件| 国产高清有码在线观看视频| 国产午夜福利久久久久久| 欧美 日韩 精品 国产| 午夜激情久久久久久久| 日韩,欧美,国产一区二区三区| 亚洲国产最新在线播放| 黄片无遮挡物在线观看| 在线观看人妻少妇| 久久久久久久精品精品| 午夜福利网站1000一区二区三区| 中文字幕亚洲精品专区| 男女边吃奶边做爰视频| 在线a可以看的网站| 少妇熟女欧美另类| 爱豆传媒免费全集在线观看| 亚洲最大成人中文| 亚洲精品日本国产第一区| 又大又黄又爽视频免费| av专区在线播放| 一区二区三区免费毛片| 亚洲av国产av综合av卡| 免费黄频网站在线观看国产| 深爱激情五月婷婷| 永久免费av网站大全| 特大巨黑吊av在线直播| 久久精品综合一区二区三区| av福利片在线观看| 干丝袜人妻中文字幕| 亚洲国产欧美在线一区| 九草在线视频观看| 亚洲精品成人av观看孕妇| 国产精品久久久久久精品电影| 欧美+日韩+精品| 国产亚洲精品久久久com| 看非洲黑人一级黄片| 又黄又爽又刺激的免费视频.| 国产国拍精品亚洲av在线观看| 国产探花在线观看一区二区| 亚洲av国产av综合av卡| 国产国拍精品亚洲av在线观看| 又黄又爽又刺激的免费视频.| 啦啦啦在线观看免费高清www| a级一级毛片免费在线观看| 久久人人爽人人爽人人片va| 亚洲人成网站在线观看播放| 丝袜美腿在线中文| 天天躁夜夜躁狠狠久久av| 伊人久久国产一区二区| 欧美一区二区亚洲| 91久久精品电影网| 伊人久久国产一区二区| 日本av手机在线免费观看| 91久久精品电影网| 尤物成人国产欧美一区二区三区| 免费大片18禁| 在线看a的网站| 日日啪夜夜爽| 国产高清国产精品国产三级 | 日本午夜av视频| 波多野结衣巨乳人妻| 久久久久九九精品影院|