Complete self-shrinkers confined into some regions of the space
aa r X i v : . [ m a t h . DG ] D ec COMPLETE SELF–SHRINKERS CONFINED INTO SOMEREGIONS OF THE SPACE
STEFANO PIGOLA AND MICHELE RIMOLDI
Abstract.
We study geometric properties of complete non–compactbounded self–shrinkers and obtain natural restrictions that force thesehypersurfaces to be compact. Furthermore, we observe that, to a cer-tain extent, complete self–shrinkers intersect transversally a hyperplanethrough the origin. When such an intersection is compact, we deducespectral information on the natural drifted Laplacian associated to theself–shrinker. These results go in the direction of verifying the validityof a conjecture by H. D. Cao concerning the polynomial volume growthof complete self–shrinkers. A finite strong maximum principle in casethe self–shrinker is confined into a cylindrical product is also presented.
Contents
Introduction 11. Some notations 42. A maximum principle 53. Self–shrinkers in a ball 73.1. Estimate of the exterior radius 73.2. Bounded self–shrinkers with | A | ≤ | A | < | A | ∈ L p ≥ m Introduction
By a self shrinker “based at” x ∈ R m +1 we mean a connected, isometri-cally immersed hypersurface x : Σ m → R m +1 whose mean curvature vector Date : November 13, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Bounded self–shrinkers, hyperplane intersection, weightedmanifolds, drifted Laplacian. field H satisfies the equation ( x − x ) ⊥ = − H , where ( · ) ⊥ denotes the projection on the normal bundle of Σ. Note that weare using the convention H = tr Σ A , where the second fundamental form of the immersion is defined as the gen-eralized Hessian A = Ddx.
With this convention, if Σ is oriented by the outer unit normal ν and we let H = Hν, then Σ is mean–convex provided H ≤ h x − x , ν i = − H. In this paper we shall consider only self–shrinkers based at 0 ∈ R m +1 .Natural examples of complete, properly embedded self-shrinkers are thecylindrical products(0.1) C k,m − k √ k = S k √ k × R m − k , k = 0 , ..., m, which include, as extreme cases, the sphere S m √ m and all the hyperplanesthrough the origin of R m +1 . Actually, according to a classification theoremby T. Colding and W. Minicozzi, [8], these are the only complete, embeddedand mean-convex self-shrinkers with extrinsic polynomial volume growth,i.e., vol( B m +1 R ∩ Σ) ≤ CR n for some C > n ∈ N and for every R >>
1; here B m +1 R denotes the ballin the ambient Euclidean space.We stress that it was conjectured by H.-D. Cao, [5], that every completeself–shrinker has extrinsic polynomial (Euclidean, in fact) volume growth.By a very interesting result due to X. Cheng and D. Zhou, [7], that com-pletes a previous theorem by Q. Ding and Y.L. Xin, [9], this is equivalent tothe fact that the immersion is proper. Thus, by way of example, if Cao Con-jecture was true, then any complete self–shrinker in a ball of R m +1 should becompact. In order to obtain indications on the validity of this conjecture, itis then relevant to understand which geometric constraints are imposed bythe assumption that a complete self–shrinker is bounded and to obtain nat-ural and general restrictions that force these hypersurfaces to be compact.For instance, we will prove the following results. Theorem 1.
Let x : Σ m → B m +1 R (0) ⊂ R m +1 be a complete self–shrinker. (a) Assume | A | ≤ . Then: (a.1) R ≥ sup Σ | H | = √ m . OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 3 (a.2) If m = 2 , then Σ = S √ . (a.3) If m ≥ and Σ is non-compact, then Σ must be connectedat infinity, i.e., it has only one end. Moreover, | A | < , theuniversal cover ˜Σ enjoys the loops to infinity property alongevery ray, [21] , and every f.g. subgroup of the fundamental groupof Σ grows at most polynomially of order m . (b) Assume lim R →∞ sup Σ \ B Σ R | A | < . Then Σ is compact. (c) Assume | A | ∈ L p (Σ) , for some p ≥ m . Then Σ is compact. More generally, one can try to understand the geometry of self–shrinkerswhich are confined in a connected region bounded by some dilated cylinder C k,m − kR , R ≥ √ k . In this setting, as a preliminary and simple fact, weobserve the validity of the following (finite) strong maximum principle. Theorem 2.
Let x : Σ m → R m +1 be a complete self-shrinker. Assume that | H | ≤ √ k and that x (Σ) is confined inside the domain bounded by C k,m − kR .If x (Σ) ∩ C k,m − kR = ∅ then: (a) R = √ k , (b) x : Σ → C k,m − k √ k is a Riemannian covering map. In particular, if k ≥ , then Σ = C k,m − k √ k in the Riemannian sense. Actually, when k = 0 and, hence, C ,m is a hyperplane through the origin,it is reasonable to expect that the self–shrinker cannot be located into oneof the corresponding half-spaces. We are able to verify that, to a certainextent, this is in fact true. The next result can be considered as a weakhalf–space theorem for complete self–shrinkers. Theorem 3.
Let x : Σ m → R m +1 be a complete, self–shrinker. Assume thateither one of the following assumptions is satisfied: (a) Σ has extrinsic polynomial volume growth (equivalently, Σ is properlyimmersed). (b) | A | ∈ L p ( d vol f ) with | A | ≤ p , for some p > .Then, for every hyperplane Π through the origin of R m +1 , Σ cannot be con-tained in one of the closed half–spaces determined by Π unless Σ = Π . Accordingly, and in view of the strong maximum principle, it is also rea-sonable to assume that some transversal intersection between a self–shrinkerand a hyperplane through the origin occurs. When such an intersection iscompact, we can obtain information on the spectrum of the natural driftedLaplacian ∆ f = ∆ − h∇ , ∇ f i , with f = | x | / Theorem 4.
Let i : Σ m ֒ → R m +1 be a complete, embedded self-shrinker.Assume that, for some hyperplane Π ≈ R m through the origin, Σ ∩ Π = K is a compact ( m − -dimensional submanifold. Then: (a) for every connected component Σ of Σ \ K (which is an open sub-manifold Σ ⊂ Σ with ∂ Σ ⊆ K ) it holds λ ( − ∆ Σ f ) ≥ . STEFANO PIGOLA AND MICHELE RIMOLDI (b)
If either Σ is compact or Σ has only one end, then there exists acompact connected component Σ of Σ \ K such that λ ( − ∆ Σ f ) = 1 . (c) If Σ is an end of Σ with respect to K and vol (cid:0) Σ ∩ B m +1 R (cid:1) = O ( e αR ) , as R → + ∞ , for some ≤ α < / , then λ ( − ∆ Σ f ) = 1 . Acknowledgments.
The authors would like to thank Pacelli Bessa, DeboraImpera and Giona Veronelli for their interest in this work and for severalsuggestions that have improved the presentation of the paper.1.
Some notations
Throughout the paper we let f = | x | d vol f the corresponding weighted volume measure of Σ,i.e., d vol f = e − f d vol . Thus, Σ f = (Σ , g, d vol f ) is a smooth metric measure space. The weightedmeasure of the intrinsic geodesic ball B Σ R ( o ) = { p ∈ Σ : d Σ ( o, p ) < R } isgiven by vol f (cid:0) B Σ R (cid:1) = Z B Σ R d vol f . Note that, obviously,vol f (cid:0) B Σ R ( o ) (cid:1) ≤ vol f ( B R ( x ( o )) ∩ Σ) , where B R denotes the Euclidean ball.There is a natural drifted Laplacian on Σ f defined by∆ f = e f div (cid:16) e − f ∇ (cid:17) = ∆ − h∇ , ∇ f i . It is symmetric on L ( d vol f ) and it can be expressed in the equivalent form∆ x T = ∆ − (cid:10) ∇ , x T (cid:11) , where x T denotes the tangential component of the immersion.Recall also that the Bakry-Emery Ricci tensor of Σ f is defined byRic f = Ric + Hess ( f ) . Using once again the self–shrinker equation we easily obtain the followingvery important estimate, [19],(1.1) Ric f ≥ − | A | where A denotes the second fundamental tensor of the immersion x : Σ m → R m +1 . Indeed, by Gauss equations,Ric ≥ h H , A i − | A | g, whereas, by the self-shrinker equation,Hess( f ) = g + D x ⊥ , A E = g − h H , A i . A maximum principle
To begin with, we observe that if a complete self–shrinker with | A | ≤ S m √ m . The analytic proof isa straightforward application of the maximum principle for subharmonicfunctions. Later on, in Section 3.2, we shall come back on this kind ofarguments. Proposition 5.
Let x : Σ m → R m +1 be a complete bounded self–shrinkerwith | H | ≤ √ m . If there exist x ∈ Σ such that | x | ( x ) = sup Σ | x | , then | x | ≡ √ m and Σ is the standard sphere S m √ m .Proof. Recall that, [8],(2.1) ∆ | x | = 2 (cid:16) m − | H | (cid:17) , therefore, by assumption, ∆ | x | ≥ . Using the strong maximum principle we thus obtain | x | ≡ c >
0. Thisimplies that x : Σ m → S mc is a Riemannian covering projection, hence anisometry since S mc is simply connected. In particular, by the self–shrinkerequation, c = √ m . (cid:3) The above result can be deduced more geometrically via a suitable appli-cation of the usual touching principle. We adopt this viewpoint to obtainthe following strong maximum principle for self-shrinkers. Recall that theoriented hypersurface x : Σ m → R m +1 is called mean–convex at p ∈ Σ if H ( p ) = H ( p ) ν where H ( p ) ≤ ν is the outward pointing unit normalat p . Theorem 6 (Maximum principle) . Let Ω ⊂ R m +1 be a domain such that i : ∂ Ω ֒ → R m +1 is a properly embedded self-shrinker. Let x : Σ m → R m +1 be a complete self-shrinker satisfying x (Σ) ⊆ Ω λ for some λ > , where Ω λ = λ Ω denotes the λ -dilation of Ω . Assume that x (Σ) ∩ ∂ Ω λ = ∅ andthat, for each intersection point x ( p ) , there exist a neighborhood V ⊂ R m +1 of x ( p ) and a neighborhood W ⊂ Σ of p such that: (i) ∂ Ω ∩ λ − V is mean convex (ii) sup W | H Σ | ≤ inf λ − V ∩ ∂ Ω | H ∂ Ω | .Then STEFANO PIGOLA AND MICHELE RIMOLDI (a) λ = 1 , (b) ∂ Ω = S k √ k × R m − k , for some k ∈ { , ..., m } , (c) x : Σ → ∂ Ω is a Riemannian covering map.In particular, if ∂ Ω is simply connected (e.g. if k ≥ in (b)), then Σ = ∂ Ω in the Riemannian sense. A situation of special interest is obtained by choosing ∂ Ω to be a cylin-drical product shrinker C k,m − k √ k . Note that the case k = m is precisely thecontent of Proposition 5. Corollary 7.
Let x : Σ m → R m +1 be a complete self-shrinker. Assume that | H Σ | ≤ √ k and that x (Σ) is confined inside the solid cylinder bounded by C k,m − kR = S kR × R m − k . If x (Σ) ∩ C k,m − kR = ∅ then (a) R = √ k , (b) x : Σ → C k,m − k √ k is a Riemannian covering map.In particular, if k ≥ , then Σ = C k,m − k √ k .Proof (of Theorem 6). Let O = x − ( ∂ Ω λ ) . Since x is smooth and ∂ Ω λ is closed in R m +1 , we have that O is a closedsubset of Σ. We claim that O is also open so that, by a connectednessargument, O = Σ i.e. x (Σ) ⊆ ∂ Ω λ . To this end, let p ∈ O . Observethat, by the mean-convexity assumption (i), in a connected neighborhood λ − U x ( p ) ⊂ ∂ Ω it holds H ∂ Ω = H ∂ Ω ν ∂ Ω , where H ∂ Ω ≤ ν ∂ Ω denotes the exterior pointing unit normal to ∂ Ω.Moreover, the rescaling property of the mean curvature tells us that H ∂ Ω λ ( x ( p )) = λ − H ∂ Ω ( λ − x ( p )) . Whence, using the fact that i : ∂ Ω ֒ → R m +1 is a self-shrinker, it is standardto deduce that either H ∂ Ω λ ≡ U x ( p ) , or H ∂ Ω λ < U x ( p ) ; see e.g. thebeginning of the proof of [19, Theorem 2]. In the first case, by assumption,we must have H Σ = 0 in a neighborhood of p in Σ and the result reducesto a well known local maximum principle for minimal surfaces. Therefore,from now on, we assume H ∂ Ω λ < U x ( p ) . Since x (Σ) lies inside Ω λ , then x (Σ) must intersect ∂ Ω λ tangentially at p ∈ O and ν Σ ( p ) = ν ∂ Ω λ ( x ( p ))the outward pointing unit normal to Ω λ . It follows from the self-shrinkerequations for ∂ Ω and Σ, and the rescaling property of the mean curvature,that H Σ ( p ) = λ H ∂ Ω λ ( x ( p )) = λ H ∂ Ω ( λ − x ( p )) . OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 7
Combining this latter with assumption (ii) we get λ | H ∂ Ω λ ( x ( p )) | = | H Σ ( p ) | ≤ (cid:12)(cid:12) H ∂ Ω (cid:0) λ − x ( p ) (cid:1)(cid:12)(cid:12) = λ | H ∂ Ω λ ( x ( p )) | . Thus λ ≤ . If we write, in a neighborhood of p : H Σ = H Σ ν Σ and H ∂ Ω λ = H ∂ Ω λ ν ∂ Ω λ , then, by mean convexity of U x ( p ) , by the above equation at p , and by con-tinuity, we have, in a neighborhood of p , H Σ , H ∂ Ω λ ( x ) < H Σ ≥ H ∂ Ω (cid:0) λ − x (cid:1) = λH ∂ Ω λ ( x ) ≥ H ∂ Ω λ ( x ) . We can now apply the usual touching principle and deduce that, actually, x (Σ) and ∂ Ω λ coincide in a small neighborhood of p . This proves the claimand, as already remarked at the beginning of the proof, x (Σ) ⊆ ∂ Ω λ .Now, x : Σ → ∂ Ω λ is a local isometry between complete manifolds, hence,it is a covering map. In particular, x (Σ) = ∂ Ω λ , and from the equality H Σ ( p ) = λ H ∂ Ω λ ( x ( p ))we deduce H ∂ Ω λ ( x ) = H Σ = λ H ∂ Ω λ ( x ) , that is λ = 1 . This shows that x (Σ) = ∂ Ω. Finally, by assumption (i), ∂ Ω is a properlyembedded self-shrinker satisfying H ∂ Ω ≤ (cid:3) Self–shrinkers in a ball
The aim of this section is to show that certain boundedness conditions onthe norm of the second fundamental form prevent the existence of complete,non–compact, bounded self–shrinkers.3.1.
Estimate of the exterior radius.
The sphere S m √ m is a self–shrinkerof constant mean curvature −√ m and contained in the compact ball B m +1 √ m (0).Our first remark is that if a complete self–shrinker with controlled intrinsicvolume growth is contained in some ball B m +1 R (0) , then there is an obviousrelation between the ray R and the dimension m . STEFANO PIGOLA AND MICHELE RIMOLDI
Proposition 8.
Let x : Σ m → R m +1 be a complete non–compact self–shrinkerwhose intrinsic volume growth satisfies R → R log vol (cid:0) B Σ R (cid:1) L (+ ∞ ) . If x (Σ) ⊆ B m +1 R (0) , then R ≥ sup Σ | H | ≥ √ m. Proof.
Recall that, by the self–shrinker equation,∆ f | x | = 2 (cid:0) m − | x | (cid:1) . On the other hand, since c − d vol f ≤ d vol ≤ cd vol f for a large enough constant c >
1, then R log vol f (cid:0) B Σ R (cid:1) L (+ ∞ )and this implies that the weighted manifold Σ f enjoys the weak maximumprinciple at infinity for the drifted Laplacian ∆ f , [15, 16]. Therefore0 ≥ (cid:18) m − sup Σ | x | (cid:19) ≥ (cid:0) m − R (cid:1) , and the claimed lower estimate on R follows. Now, from the self–shrinkerequation we have sup Σ | H | ≤ | x | ≤ R . Using this information into equation (2.1):∆ | x | = 2 (cid:16) m − | H | (cid:17) , and noting also that the weak maximum principle at infinity for the Lapla-cian holds on Σ, we deduce0 ≥ (cid:18) m − sup Σ | H | (cid:19) . This completes the proof. (cid:3)
Remark 9.
In particular, if Σ has extrinsic polynomial volume growth, thenthe above radius estimate holds. This follows from the obvious relationvol( B Σ R ) ≤ vol( B m +1 R ∩ Σ) . Note that, by [23, Theorem 2.2] and inequality (1.1), a complete non–compact bounded self–shrinker x : Σ m → R m +1 with | A | ≤ B Σ R ) ≤ CR m . OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 9
Moreover, since | A | ≤
1, by the Cauchy–Schwarz inequality we have that | H | ≤ m . We can hence specialize Proposition 8 to the following Corollary 10.
Let x : Σ m → R m +1 be a complete non–compact self–shrinker with | A | ≤ . If x (Σ) ⊆ B m +1 R (0) , then R ≥ sup Σ | H | = √ m. Bounded self–shrinkers with | A | ≤ . As a consequence of thestrong maximum principle for the Laplace-Beltrami operator, we observedin Section 2 that, for a self–shrinker satisfying | A | ≤
1, hence | H | ≤ √ m , thenorm of the immersion cannot attain a finite maximum unless the shrinkeris a round sphere of radius √ m . In particular, this applies to any compactself–shrinker with the same bound on the mean curvature. It is by nowwell understood that parabolicity is a good substitute of compactness. Fortwo-dimensional shrinkers this property is implied by the above conditionon the second fundamental form. Theorem 11.
Let x : Σ → R be a complete bounded self–shrinker with | A | ≤ . Then Σ = S √ .Proof. Since m = 2, we know from (3.1) that Σ has quadratic intrinsicvolume growth, therefore it is parabolic (possibly compact); see e.g. [10].As in Proposition 5, since | H | ≤ √ | x | is a bounded subharmonic functionand we obtain that | x | ≡ const . This implies Σ = S √ . (cid:3) In higher dimensions, the same control gives information on the topologyat infinity of a bounded shrinker.
Theorem 12.
Let x : Σ m → R m +1 be a complete non–compact boundedself–shrinker with | A | ≤ . Then Σ does not contain a line. In particular, Σ is connected at infinity, i.e., Σ has only one end. Remark 13.
Applying this result to the universal covering of Σ, and using[21, 22, 23], we also get the topological information collected in Theorem 1stated in the Introduction.
Proof.
Assume by contradiction that Σ contains a line. By assumption and(1.1), we have that
Ric f ≥ f bounded. Therefore, we can apply theCheeger–Gromoll–Lichnerowicz splitting theorem, [12], and obtain that Σsplits isometrically as the Riemannian product (cid:0) N m − × R , g N + dt ⊗ dt (cid:1) .Moreover f is constant along the line. Thus(3.2) Hess( f )( ∂ t , ∂ t ) = 0 . On the other hand, consider the Simons type equation, see e.g. [11, 8],(3.3) 12 ∆ f | A | + | A | (cid:16) | A | − (cid:17) = | D A | . Since, by assumption, | A | ≤ | A | < | A | ≡
1, on Σ. In case(a), recalling (1.1), we deduce thatRic f ( ∂ t , ∂ t ) = Hess( f )( ∂ t , ∂ t ) > , contradicting (3.2). Suppose that (b) holds, namely, | A | ≡
1. Using againthe Simons equation we get that A is parallel. We can therefore applya classification theorem by Lawson and deduce that x (Σ) is a cylindricalproduct S k √ k × R m − k with k = 0 , ..., m . Since the self–shrinker is bounded,we conclude that Σ = S m √ m , contradicting the assumption that Σ is notcompact. (cid:3) Bounded self–shrinkers with lim sup | A | < . In the two previousresults we considered global bounds on the norm of the second fundamentalform. The application of the Feller property for ∆ f in combination with themaximum principle at infinity enable us to prevent the existence of complete,non–compact, bounded self–shrinkers even in the case a pinching conditionon | A | is required at infinity. Recall that the weighted manifold Σ f is saidto be Feller if, for some (hence any) smooth domain Ω ⊂⊂ Σ f and λ > h > (cid:26) ∆ f h = λh on Σ \ Ω h = 1 on ∂ Ωsatisfies h ( x ) → x → ∞ ; see [18, 3]. In particular we obtain thefollowing Theorem 14.
Let x : Σ m → B m +1 R (0) ⊂ R m +1 be a complete self–shrinkerwith lim R →∞ sup Σ \ B Σ R | A | < . Then Σ is compact. Remark 15.
Suppose that Σ is compact. Then Σ \ B Σ R = ∅ for R > diam(Σ)and, therefore, lim R →∞ sup Σ \ B Σ R | A | = −∞ , proving that the assumptionof the theorem is automatically satisfied. Note also that, from a differentperspective, the result states that a complete, non–compact, bounded self–shrinker must satisfy the asymptotic condition lim R →∞ sup Σ \ B Σ R | A | ≥ Proof.
First observe that, since | A | ∈ L ∞ (Σ) and |∇ f | = | x T | ≤ | x | OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 11 On the other hand, using the self-shrinker equation, we computeHess ( f ) = g − h H , A i . By (3.5) having fixed any ray γ : [0 , + ∞ ) → Σ, we have d dt ( f ◦ γ ) ( t ) = Hess ( f ) ( ˙ γ, ˙ γ ) ≥ 12 , for t >> . It follows by integration that | x | → + ∞ along γ and, therefore, x (Σ) isunbounded. Contradiction. (cid:3) Bounded self–shrinkers with | A | ∈ L p ≥ m . In the next result weswitch from L ∞ to L p conditions on the norm of the second fundamentalform. In particular we show that complete bounded self–shrinkers with finitetotal curvature must be compact. Theorem 16. Let x : Σ m → R m +1 be a complete, bounded self–shrinkersatisfying | A | ∈ L p ( d vol) , for some p ≥ m . Then Σ is compact.Proof. By contradiction, suppose that Σ is complete and non-compact. Toillustrate the argument, let us first consider the case p = m . Since f is bounded and | H | ∈ L m (Σ) it is standard to obtain that Σ enjoys theweighted L -Sobolev inequality (cid:18)Z ϕ mm − d vol f (cid:19) m − m ≤ S Z |∇ ϕ | d vol f , for some constant S > ϕ ∈ C ∞ c (Σ). Indeed, first we canabsorb the mean curvature term in the Sobolev inequality by J.H. Michaeland L.M. Simon, [13], outside a large compact set, then, according to [4],we can extend the resulting Sobolev inequality to all of Σ and, finally, wenote that, since f is bounded, c − d vol f ≤ d vol ≤ c d vol f for a large enough constant c > f | A | + | A | ≥ . Since | A | ∈ L m ( d vol f ), combining the PDE with the weighted Sobolevinequality gives the Anderson-type decay estimate(3.6) sup Σ \ B Σ R ( o ) | A | = o (cid:0) R − (cid:1) , as R → + ∞ . This follows e.g. by adapting to the weighted setting the arguments in [14].From this uniform estimate it is now standard to get that the immersion x is proper, thus contradicting the assumption that x (Σ) is a bounded subsetof R m +1 . In fact, we have the following general result that, in the setting ofminimal submanifolds of the Euclidean space, traces back to a paper by M.Anderson, [1]; see also Remark 18 below. Lemma 17. Let x : (Σ m , g ) → R m +1 be a complete, non-compact hypersur-face satisfying (3.6). Then x is proper and Σ has finite topological type, i.e.,there exists a smooth compact subset Ω ⊂⊂ Σ such that Σ \ Ω is diffeomorphicto the half-cylinder ∂ Ω × [0 , + ∞ ) . As a matter of fact, the uniform decay condition (3.6) on the secondfundamental form, as well as the corresponding structure Lemma, are eventoo much strong for the desired conclusion to hold. This is illustrated in thenext reasonings where we assume the general condition p ≥ m .Again, by contradiction, suppose that Σ is complete and non–compact.Since f is bounded, by the self–shrinker equation we get | H | ∈ L ∞ . Whence,we obtain that Σ enjoys the weighted L -Sobolev inequality (with potentialterm) (cid:18)Z Σ ϕ mm − d vol f (cid:19) m − m ≤ A Z Σ |∇ ϕ | d vol f + B Z Σ ϕ d vol f , for some constants A, B > ϕ ∈ C ∞ c (Σ). Since | A | is asolution of the semilinear equation∆ f | A | + | A | ≥ , and | A | ∈ L p ( d vol f ) = L p ( d vol) for some p ≥ m , we deduce that (see e.g.[14])(3.7) sup Σ \ B Σ R | A | = o (1) , as R → + ∞ . Reasoning exactly as in the last part of the proof of Theorem 14 this leadsto the fact that x (Σ) is unbounded, yielding a contradiction. (cid:3) Remark 18. The decay assumption (3.6) in Lemma 17 can be considerablyrelaxed. This was established in [2] where the authors used the notion oftamed submanifolds. We are grateful to Pacelli Bessa for having pointedout this fact to us.4. Self–shrinkers and hyperplanes through the origin Self–shrinkers in a half–space. It is reasonable that a completeself–shrinker has a certain homogeneous distribution around 0 ∈ R m +1 and,therefore, it should intersect every hyperplane through the origin. For com-pact self–shrinkers this property is easily verified. In fact, more is true. Itwas proved in Theorem 7.3 of [22] that if the distance between two prop-erly immersed self–shrinkers (either compact or not) is realized, then theself–shrinkers must intersect. In particular, a compact self–shrinker mustintersect every hyperplane through the origin, as claimed. Moreover, theintersection must be non-tangential by maximum principle considerations.Summarizing, a compact self–shrinker cannot be contained in one of thehalf–spaces determined by a hyperplane through the origin. Needless to say,exactly the same proof works for a complete self–shrinker with polynomial OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 13 volume growth because, according to [7], it is properly immersed. We aregoing to recover the same conclusion by using more direct and analytic argu-ments that are suitable for a generalization to the complete, (non–necessarilyproper) setting. Theorem 19. Let x : Σ m → R m +1 be a compact self–shrinker. Then, forevery hyperplane Π through the origin of R m +1 , x (Σ) cannot be containedin one of the closed halfspaces determined by Π .Proof. Recall that, for a self–shrinker,∆ f x = − x. Therefore, if Π has normal equation(4.1) Π : L ( y ) := m +1 X j =1 a j y j = 0 , we have that the self–shrinker satisfies also(4.2) ∆ f L ( x ) = − L ( x ) . Whence, it follows easily that x (Σ) cannot be contained in one of the closedhalf-spaces determined by Π. Indeed, otherwise, we would have that either L ( x ) ≥ L ( x ) ≤ 0. Without loss of generality, suppose that L ( x ) ≥ L ( x ) would be an f -superharmonic functionon the compact manifold Σ. By the maximum principle L ≡ const andby equation (4.2) L ≡ 0. This means that x (Σ) ⊆ Π and, by geodesiccompleteness, x (Σ) = Π. This is clearly impossible because Σ is compact. (cid:3) A similar conclusion can be obtained for complete self–shrinkers x : Σ m → R m +1 with a controlled extrinsic geometry. By way of example, suppose that(4.3) | x | + | A ( p ) | ≤ q r ( p ) , where r ( p ) = d Σ ( p, o ). Then, for every hyperplane Π through the origin, if x (Σ) lies on one side of Π, thendist R m +1 (Π , x (Σ)) = 0and the distance is not attained, unless x (Σ) = Π.Indeed, note that, in light of (1.1), condition (4.3) implies Ric f ≥ − C (1 + r ), |∇ f | = | x T | ≤ p r . Then, according to Corollary 5.3 in [17], for every u ∈ C (Σ) with inf Σ u = u ∗ > −∞ there exists a sequence { p n } ⊂ Σ along which u ( p n ) < u ∗ + 1 n , |∇ u | ( p n ) < n , ∆ f u ( p n ) > − n . Now, as in the compact case, if x (Σ) lies on one side of Π, we can as-sume that L ( x ) ≥ L ( y ) is defined in (4.1). Evaluating (4.2) along { p n } we deduce that inf Σ L ( x ) = 0, as desired. The second conclusion is aconsequence of the strong minimum principle for positive super-solutions of∆ f + 1.In the next theorem we point out natural geometric conditions that permitto recover the full conclusion of the compact case. Theorem 20. Let x : Σ m → R m +1 be a complete, non-compact self–shrinker.Assume that either one of the following assumptions is satisfied: (a) Σ has (extrinsic) polynomial volume growth. (b) vol f ( B Σ R ) = O ( R ) as R → ∞ . (c) | A | ∈ L p ( d vol f ) and | A | ≤ p , for some p > .Then, for every hyperplane Π through the origin, if x (Σ) lies on one side of Π , then x (Σ) = Π .Proof. We shall use extensively the notation introduced so far. In particular,the hyperplane Π is described by the normal equation (4.1) and the function L ( x ) satisfies equation (4.2).Assume we are in the assumptions of (a) . Since Σ has polynomial volumegrowth, then vol f (Σ) < + ∞ and Σ f is parabolic with respect to the driftedLaplacian ∆ f . Using the above notation, assume without loss of generalitythat L ( x ) ≥ 0. By equation (4.2) we see that L ( x ) ≥ f -superharmonic,hence it is constant by f –parabolicity. The desired conclusion now followsas in the proof of Theorem 19. Case (b) is completely similar. Assumenow that the assumptions in (c) are satisfied. Let x (Σ) = Π and, bycontradiction, suppose that x (Σ) is contained in a half-space determined byΠ. Then, by the strong minimum principle, we can assume that L ( x ) > f L + L = 0 . Since p (cid:16) | A | − (cid:17) ≤ , for some p > 1, we obtain∆ f L + p (cid:16) | A | − (cid:17) L ≤ . Combining this latter with the Simons–type inequality | A | n ∆ f | A | + | A | (cid:16) | A | − (cid:17)o ≥ | D A | − |∇ | A || ≥ , and applying Theorem 8 in [19] we conclude that either | A | ≡ | A | ≡ f | A | + | A | (cid:16) | A | − (cid:17) = | D A | gives that | D A | ≡ x (Σ) = S k √ k × R m − k , with 0 ≤ k ≤ m . Since x (Σ) must lie on one side of Π we necessarilyhave k = 0, i.e., x (Σ) = Π, contradiction. (cid:3) OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 15 Bottom of the spectrum of the drifted Laplacian. Once wehave understood that, to a certain extent, complete self–shrinkers inter-sect transversally a hyperplane through the origin, we are going to deducespectral information on the drifted Laplacian whenever the intersection iscompact, and some (extrinsic) volume growth condition is satisfied.The intuition for the general result contained in Theorem 23 relies onthe following two examples. Recall that, by definition, the bottom of thespectrum of − ∆ f on a domain Ω ⊆ Σ, with Dirichlet boundary conditions,is defined by λ ( − ∆ Ω f ) = inf v ∈ C ∞ c (Ω) \{ } R Ω |∇ v | d vol f R Ω v d vol f . The bottom of the spectrum λ is an eigenvalue of − ∆ f if there exists afunction u ∈ Dom( − ∆ Ω f ) such that − ∆ f u = λ u on Ω , where Dom( − ∆ Ω f ) = { u ∈ W , (Ω , d vol f ) : ∆ f u ∈ L (Ω , d vol f ) } is the domain of (the Friedrichs extension of) − ∆ f originally defined on C ∞ c (Ω). For future purposes, we also recall that if ∂ Ω is compact then, u ∈ W , (Ω , d vol f ) and u = 0 on ∂ Ω ⇒ u ∈ W , (Ω , d vol f ) . Indeed, the interesting case occurs when Ω is non–compact, i.e., an exteriordomain, in the complete manifold Σ. Let 0 ≤ φ R ≤ B Σ2 R , satisfying φ R = 1 on B Σ R andsuch that |∇ φ R | ≤ /R . Then, u R = uφ R ∈ W , (Ω) and it is easy to verifythat u R → u in W , (Ω , d vol f ), as R → ∞ . Example 21. Consider the self–shrinker sphere S m √ m . Then, each hy-perplane Π through the origin divides S m √ m into half–spheres isometric to + S m √ m = S m √ m ∩ (cid:8) y m +1 > (cid:9) . Since f ( x ) ≡ m/ 2, it holds λ ( − ∆ + S m √ m f ) = λ ( − ∆ + S m √ m ) = 1 m λ ( − ∆ + S m ) = 1;see e.g. [6]. Example 22. Consider the self–shrinker cylinder C = S m − √ m − × R . Thenthe hyperplane Π = (cid:8) y m +1 = 0 (cid:9) intersects C along the sphere S m − √ m − anddivides C into two half–cylinders isometric to C + = S m − √ m − × R + . These arethe ends of Σ. We claim that λ ( − ∆ C + f ) = 1 . Indeed, since f = | x | m − 12 + x m +1 , we have the decomposition∆ C + f = ∆ S m − √ m − + ∆ R + t / and, therefore, λ ( − ∆ C + f ) = λ ( − ∆ S m − √ m − ) + λ ( − ∆ R + t / )= 0 + λ ( − ∆ R + t / ) . Now, the Ornstein–Uhlenbeck operator ∆ R t / on ( R + , e − t / dt ) satisfies λ ( − ∆ R + t / ) = 1 . See e.g. the lecture notes [20] for the basic theory and more advanced topicson the Ornstein–Uhlenbeck operator and its semigroup. Indeed, u ( t ) = t isa smooth, positive function on R + satisfying(4.4) ∆ R + t / u = u ′′ − tu ′ = − u so that, by (the weighted version of) Barta’s theorem, λ (∆ R + t / ) ≥ inf R + − ∆ R + t / uu = 1 . On the other hand, u ∈ W , ( R + , e − t / dt ), therefore, by (4.4), ∆ R + t / u ∈ L ( R + , e − t / dt ). Furthermore, u (0) = 0. It follows that u ∈ Dom( − ∆ R + f )is also a Dirichlet eigenfunction of the Ornstein–Uhlenbeck operator on R + .Abstracting from the previous examples we are now ready to state thefollowing general result. Theorem 23. Let i : Σ m ֒ → R m +1 be a complete, embedded self–shrinker.Assume that, for some hyperplane Π ≈ R m through the origin, Σ ∩ Π = K is a compact ( m − –dimensional submanifold. Then: (a) for every connected component Σ of Σ \ K (which is an open sub-manifold Σ ⊂ Σ with ∂ Σ ⊆ K ) it holds λ ( − ∆ Σ f ) ≥ . (b) If either Σ is compact or Σ has only one end, then there exists abounded connected component Σ of Σ \ K such that λ ( − ∆ Σ f ) = 1 . (c) If Σ is an end of Σ with respect to K with extrinsic volume growth (4.5) vol (cid:0) Σ ∩ B m +1 R (cid:1) = O ( e αR ) , as R → + ∞ , for some ≤ α < / , then λ ( − ∆ Σ f ) = 1 . OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 17 Remark 24. The conclusion in (a) holds regardless of the fact that theintersection K is compact. Remark 25. Note that condition (4.5) in (c) is actually equivalent to the(only apparently less general) polynomial volume growth condition. Indeed,it is easy to see that (4.5) implies that vol f (Σ ) < + ∞ (see Lemma 26 below)and minor changes to the proofs of Theorem 2.1 and Theorem 4.1 in [7] showthat the equivalences in [7] can be localized to a given end. In particular,under assumption (4.5), Σ is proper and of extrinsic polynomial (Euclidean)volume growth. For the sake of completeness we sketch out here the proofof the fact that a properly immersed end has Euclidean volume growth. Theproofs of the remaining implications can be easily adapted from the originalones. Suppose that ˜Σ is a properly immersed end of a complete noncompactself-shrinker x : Σ m → R m +1 . To prove that ˜Σ must have Euclidean extrinsicvolume growth observe that, since ∂ ˜Σ is compact and properly immersed wecan find a regular value r such that n p ∈ ˜Σ : | x ( p ) | = r o does not intersect ∂ ˜Σ. Then we can define for r > r the set D r := n p ∈ ˜Σ : r < | x ( p ) | < r o .Since the immersion is proper, letting h = | x | , we can define for t > r > r , I ( t ) = 1 t m Z D r e − ht d vol . Since on a self-shrinker |∇ h | − h ≤ h h + h = m, we gain that, if t ≥ I ′ ( t ) ≤ − t − m − Z D r div (cid:16) e − ht ∇ h (cid:17) . At a regular value r of | x | , for t ≥ 1, by Stokes’ Theorem we have thus I ′ ( t ) ≤ − t − m − " Z {| x | = r } (cid:28) e − ht ∇ h, ∇ h |∇ h | (cid:29) d vol − Z {| x | = r } (cid:28) e − ht ∇ h, ∇ h |∇ h | (cid:29) d vol ≤ t − m − Z {| x | = r } e − ht |∇ h | d vol . Integrating on [1 , r ], with r > r ≥ 1, and elaborating, we get(4.6) e − r − m Z D r d vol ≤ Z D r e − h d vol + Z r t − m − e − r t dt Z {| x | = r } |∇ h | d vol . Proceeding now as in [7] we can conclude that, for any positive integer N ,we have Z D r + N e − h d vol ≤ " N Y i =0 − e − ( r + i ) D r − e − h d vol+ e − r Z r t − m − e − r t r m − ( {| x | = r } ) . This implies that R ˜Σ e − h d vol < + ∞ and the desired Euclidean extrinsicvolume growth of ˜Σ follows from (4.6). Proof (of Theorem 23). Let Π be represented by the normal equationΠ : L ( y ) := m +1 X j =1 a j y j = 0 . Recall that, for every self–shrinker,∆ f x = − x. It follows that ∆ f L ( x ) + L ( x ) = 0, on Σ . In particular, this equation holds on Σ . Moreover, since Σ is contained inone of the open halfspaces determined by Π, then either L < L > . Thus, up to changing the sign of L , we can assume L > λ ( − ∆ Σ f ) ≥ inf Σ − ∆ f LL = 1 . This proves (a).Suppose now that Σ is non–compact and has only one end. We claimthat there exists a compact connected component Σ of Σ \ K . In this case,since L = 0 on ∂ Σ ⊆ K , we deduce that L is an eigenfunction of ∆ Σ f corresponding to eigenvalue +1. When combined with (a) this clearly im-plies that λ ( − ∆ Σ f ) = 1, completing the proof of (b). To prove the claim,we first observe that Σ \ K cannot be connected. Indeed, by contradiction,suppose the contrary. Then Σ must be contained in one of the closed half–spaces determined by Π and intersects Π tangentially along K . Withoutloss of generality, we can assume that L ( x ) ≥ L ( x ) = 0 on K .Since ∆ f L ( x ) = − L ( x ) ≤ L ( x ) ≡ ⊆ Π. Actually, Σ = Π by geodesic completenessand this clearly prevents K = Σ ∩ Π to be compact, contradiction. Thus,Σ \ K has at least two connected components. Since we are assuming thatΣ has one end, at most one of them can be unbounded. We therefore find abounded component Σ ⊆ Σ of Σ \ K , as claimed. OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 19 It remains to prove (c). The argument is completely similar to the above.According to (a), λ ( − ∆ Σ f ) ≥ L ( x ) ≥ (cid:26) ∆ f L ( x ) + L ( x ) = 0, on Σ L = 0, on ∂ Σ ⊆ K. To conclude that, in fact, λ ( − ∆ Σ f ) = 1 it suffices to show that L ∈ Dom(Σ ). Since L = 0 on the compact boundary ∂ Σ , we have to showthat L ∈ W , (Σ , d vol f ). To this aim, we simply note that | L ( x ) | qP a j = dist R m +1 ( x, Π) ≤ d R m +1 ( x, 0) = | x | , and |∇ L ( x ) | qP a j ≤ . Therefore, we can apply the next trivial lemma. This proves (c) and com-pletes the proof of the theorem. (cid:3) Lemma 26. Let x : Σ m → R m +1 be any hypersurface satisfying vol (cid:0) Σ ∩ B m +1 R (cid:1) = O (cid:16) e αR (cid:17) , as R → + ∞ , for some ≤ α < / . Then, for every polynomial P ( t ) and for every ≤ β < / − α , P ( | x | ) e β | x | ∈ L ( d vol f ) . Proof. Note that, by assumption, there exists t > − t α − β > . Now, we simply compute Z Σ | x | p e β | x | d vol f = Z Σ | x | p e − ( − β ) | x | d vol= C + C ∞ X n =0 Z Σ ∩ (cid:16) B m +1 tn +1 \ B m +1 tn (cid:17) | x | p e − ( − β ) | x | d vol ≤ C + C ∞ X n =0 t pn + p e − ( − β ) t n vol (cid:0) Σ ∩ B m +1 t n +1 (cid:1) ≤ C + C ∞ X n =0 t pn + p e − ( − t α − β ) t n < + ∞ . (cid:3) References [1] M. Anderson, The compactification of a minimal submanifold by its Gauss map. Preprint. [2] G. P. Bessa, L. Jorge, F. Montenegro, Complete submanifolds of R n with finite topology. Comm. Anal. Geom. (2007), 725–732.[3] G. P. Bessa, S. Pigola, A. G. 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OMPLETE SELF–SHRINKERS INTO SOME REGIONS OF THE SPACE 21 Dipartimento di Scienza e Alta Tecnologia, Universit`a degli Studi dell’Insubria,via Valleggio 11, I-22100 Como, ITALY E-mail address : [email protected] Dipartimento di Scienza e Alta Tecnologia, Universit`a degli Studi dell’Insubria,via Valleggio 11, I-22100 Como, ITALY E-mail address ::