RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION

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1 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION JINGYI CHEN AND YUXIANG LI Abstact. We show that a smooth adially symmetic solution u to the gaphic Willmoe suface equation is eithe a constant o the defining function of a half sphee in R 3. In paticula, adially symmetic entie Willmoe gaphs in R 3 must be flat. When u is a smooth adial solution ove a punctued disk Dρ\{} and is in C Dρ, we show that thee exist a constant λ and a function β in C Dρ such that u = λ log + β; moeove, the gaph of u is contained in a gaphical egion of an inveted catenoid which is uniquely detemined by λ and β. It is also shown that a adial solution on the punctued disk extends to a C function on the disk when the mean cuvatue is squae integable.. Intoduction A Willmoe suface in R 3 is a smoothly immesed suface that satisfies the equation g H + H3 H K =. whee H = κ + κ and K = κ κ ae the mean cuvatue and the Gauss cuvatue of the suface espectively and g is the Laplace-Beltami opeato in the metic g on the suface, which is induced fom the Euclidean metic in R 3 via the immesion. Fo a smooth gaph x, y, ux, y in R 3, the aea element is v = + Du and H = div Du v, whee D is the Euclidean gadient opeato on functions in the xy-plane. Fo gaphs, as shown in [4], the Willmoe suface equation. can be witten in an elegant fom: div I v Du Du DvH v H Du =.. Inspied by the classical theoem of Benstein fo the minimal suface equation, one asks when an entie Willmoe gaph is a plane. An entie smooth solution to. is shown to be affine if H is squae integable [3], [6]; the appoach taken in [3] is geometic in natue, athe than dealing with the fouth ode patial diffeential equation diectly. It has emained an inteesting open question on the adially symmetic case: is an entie adial solution to. constant? In this pape, we answe this question affimatively. Date: Octobe 3, 4. The fist autho acknowledges the patial suppot fom NSERC. The second autho is patially suppoted by NSFC. This wok was done when the second autho visits UBC in the fall of 4; he is gateful to the Depatment of Mathematics at UBC and PIMS fo poviding him a nice eseach envionment.

2 J. CHEN & Y. LI Theoem.. Let u be a smooth solution to. defined on a disk Dρ in the xyplane centeed at the oigin with adius ρ, whee = x + y. Suppose that u = c and the mean cuvatue of of the gaph x, y, u at,, c is a. Then if a =, then u c; if a, the gaph is contained in a half sphee centeed at,, c of adius a. Consequently, a global adially symmetic solution to. must be constant. The main obsevation in this pape can be summaized as follows. The fouth ode equation. educes to a thid ode equation due to its divegence fee stuctue; the latte futhe educes, in the adially symmetic case, to an odinay diffeential equation of second ode fo u since the zeoth ode tem u itself is absent fom the equation.. The singulaity at = in the second ode equation is supisingly mild. Even moe, the diffeence of any two solutions to the second ode equation satisfies an equation which admits a contaction popety that leads to uniqueness of solutions to an initial value poblem. The desied igidity then follows. In section we will pove Theoem.. In section 3, we will extend the techniques developed in section to study adially symmetic solutions with a point singulaity at the oigin. Ou main focus is to classify the adially symmetic solutions to the Willmoe suface equation on a punctued disk Dρ\{}, which is in C Dρ. Such a solution u will be shown to have the following decomposition popety: u = λ log + β, whee β C [, ρ; which is equivalent to the mean cuvatue decomposes as: H = λ log + γ, whee γ C [, ρ. Futhe, we will pove that such a suface is contained in the image of an invesion in R 3 of a catenoid, whee the invesion and the catenoid ae uniquely detemined by the infomation at the oigin: λ and β o equivalently γ. We now descibe the coesponding catenoid and invesion. A catenoid C c may be defined by the paametization F c t, θ = c cosh t cos θ, c cosh t sin θ, ct, whee c hee we allow c <. The invesion of this catenoid unde the mapping I a x, y, z = x, y, z a x, y, z a is a closed suface with R 3 as its only nonsmooth point, in paticula, away fom finitely many cicles that ae pependicula to the z-axis, the inveted catenoid can be witten as seveal gaphs ove the xy-plane, among them, we will identify which one contains the gaph of u tanslated by the constant vecto,, u. Let c cosh t ht = c cosh t + ct a

3 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION 3 be the distance of I a F c t, θ to the z-axis. Diect computations shows that h has only finitely many citical points and denote them by t < < t m. Then the invesion of C c \ i F c{t i } S consists of the following m + gaphs I a Fc, t S, I a Fc t, t S,, I a Fc t m, + S whee the fist one and the last one ae gaphs ove a punctued disk and the est ae gaphs ove an annulus. Let t c,a = t m, R c,a = ht c,a, Σ c,a = I a Fc t m, + S. The disk DR c,a in the xy-plane is the lagest domain ove which Σ c,a can be defined see Section 3. fo moe details. Theoem.. Let u be a smooth solution to. defined on a disk Dρ\{} in the xy-plane centeed at the oigin with adius ρ. Suppose u C Dρ. Then thee exist a constant λ and a function β C [, ρ such that u = λ log + β, > ; the gaph of u is contained in,, u + Σ c,a whee c = λ 4 and a = λ 4 λ log 8 + 3λ 8 β and ρ is no lage than R c,a and u C,α Dρ fo any < α <. The egulaity assumption u C Dρ in Theoem. will be shown to hold when the mean cuvatue H is squae integable ove Dρ. Coollay.3. Let u be a smooth solution to. defined on a punctued disk Dρ\{} in the xy-plane. Suppose that the Willmoe enegy Dρ H dµ is finite. Then u can be extended at as a C,α function on Dρ and and in Theoem. hold. Fo igidity of Willmoe sufaces in R 3 that ae not necessaily gaphs, Byant showed [], among othe things, that an embedded closed Willmoe suface of genus zeo must be a ound sphee; Rigoli [7] showed that fo a complete Willmoe immesion f : Σ R 3, if thee exists a vecto a R 3 such that n + Hf, a, then fσ must be a plane. We would like to thank Juncheng Wei fo binging the pape [] to ou attention afte we posted ou pape at axiv. In [], ou equation. was deived fo the socalled Helfich functional that contains thee paametes and it educes to the Willmoe functional when all thee paametes ae see also the efeence in []. Othe than this, ou esults do not ovelap with those in [] as the main theoem theein is not applicable to Willmoe sufaces.. Smooth Radially symmetic solutions Let f be a smooth adially symmetic function defined on the disk Dρ in R centeed at the oigin with adius ρ, ]. The divegence of the smooth adial vecto field

4 4 J. CHEN & Y. LI X = f, whee = x + y fo x, y R, is given by divx = f = f + f.. Moeove, f must be. In fact, if we set fx, y = f, then fx, y = f x, y. It follows that both f x and f y vanish at the oigin; hence f = f x, cos θ + f y, sin θ =. Now conside the gaph defined by a smooth adially symmetic function u on Dρ. Letting w = u, we have v = + w and v = ww v. Note that w = u =. We calculate the mean cuvatue of the gaph as follows w H = div v w w = + v v = w v w w + w v 3 v = w v + w 3 v. Two consequences ae immediate. Fist, at the oigin, we have Second, we have H = w + lim +. w v = w..3 vh = w v ww + w v 4 w..4 To simplify the divegence fee adial vecto field in the Willmoe suface equation., we fist compute I Du Du DvH = v = u x vh x u xu y v + u xu y v vh x + + u y + v u xu y v v u y v vh x u xu y v vh y vh y x vh y y x vh x + + u x vh v y y..5

5 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION 5 Next, we convet the patial deivatives in x, y of the adially symmetic functions u, vh into the deivatives in : x u x = u, u y y = u, vh x x = vh, vh y y = vh..6 It follows by substituting.6 into.5 that I Du Du DvH = v v + u y vh x + v u vh x y = x v vh = v vh. Define a adially symmetic function by Du Du f = I v v x + y xy u vh 3 x u x y 3 vh y y DvH H Du,..8 Using.,.4 and.7, we find f = v v vh H w = w v v v ww + w v 4 w v H w = w v ww + w 5 v 7 v 3 w w w + w w + w w v v 6 v v 4 = w + w v 5 v w w + v w w v w w3 v v = w 5w v 5 + w w + w + w w + w w 3 = w w 5w + v 5 + w w w3 3 + w..9 Lemma.. Let u be a smooth adially symmetic function that solves the Willmoe suface equation. on a disk Dρ in the xy-plane cented at the oigin. Then f =. Poof. Using the function defined in.8, the Willmoe suface equation. eads div f =.. By., the above equation educes to f =.

6 6 J. CHEN & Y. LI Then f = C. fo some constant C. Futhemoe, fo any < < ρ, integating the Willmoe equation. and using Stokes theoem = div f = f = π C. D D Thus, C =. Then, Lemma. and.9 yield Lemma.. F = x, y, u is Willmoe if and only if w w + ϕω =,. whee ϕω = 5w + w w + w3 3 + w. Fo., we now establish a uniqueness esult fo an initial value poblem. Lemma.3. Thee exists ɛ > such that the equation. has at most one solution on [, ɛ] with initial data w =, and w = a. Poof. Let w and w be solutions to. on [, ] fo some >. Then we have We have ϕw ϕw = = w w + w w = ϕw ϕw..3 5w + w w + w3 3 + w 5w + w w w3 3 + w 5w + w 5w w + w + 5w w + w w + w w +3 w3 w 3 + w5 w 5..4 To estimate the thee tems on the ight hand side of the above identity, we fist obseve w + w w + w = w w w w + w + w w w. Next, let Λ = w C [, ] + w C [, ]..5 Since w i =, fo, it holds w i = w sds w i C [, ].

7 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION 7 Thee exists some positive constant CΛ which depends only on Λ such that: w i CΛ, i =, ; w 3 w 3 = w + w w + w w w CΛ w w, w 5 w 5 CΛ w w. Then ϕw ϕw C w w + w w, [, ],.6 whee C depends only on Λ and. Let φ = w w. We have φ = φ =, hence lim φ + = ; theefoe by.3, Then By.6, φ + φ φ = φ C = C C = C = ϕw t ϕw tdt. ϕw t ϕw tdt. φt + tφ t dt.7 φt + tφt φt dt φt + tφt dt t sφs ds t + tφt dt 4C tφ C [,], whee, ]. Given,,, if C φ is not identically on [, ], then we can find, ], such that < φ = φ C [, ]. Thus,.7 yields < φ 4C φ < φ, which is impossible. Theefoe, φ = on [, ]. It follows that φ is a constant which must be since φ =. We conclude that φ = on [, ]. Then w = w on [, ɛ] fo ɛ = min{, C }. Poof of Theoem.: We divide the poof into two cases.

8 8 J. CHEN & Y. LI When a =, it is clea that is solution to.. Since w = u is also a solution to., by Lemma.3, w = on [, ɛ] fo some ɛ >. Ove the inteval [ɛ, +,. is a second ode odinay diffeential equation with smooth coefficients, by the standad uniqueness theoem in ODE, w = on [ɛ, + hence on [, +. Then u is a constant. Let a. It is well-known that the half sphee F = x, y, signa Willmoe suface, i.e. signa a satisfies the equation.. Moeove, signa a = a. = Simila to case, fom Lemma.3 w = signa a on [,, which is the maximum inteval on which w is defined. Then a u = signa a + c whee c is a constant and [,. a a is a Fo an entie adial solution to., the second case in Theoem. cannot occu. Theefoe, we have Coollay.4. A smooth adially symmetic Willmoe gaph F = x, y, u must be a plane that is paallel to the xy-plane. 3. Radially symmetic solutions with a point singulaity In this section, we study adially symmetic Willmoe suface which has a singulaity at,,. 3.. Regulaity and uniqueness of solutions. Examining the poof of Lemma., due to the singulaity at, we can only conclude fom. that f = λ fo some λ R, while λ = when u is smooth at. Theefoe, the equation fo w = u takes a diffeent fom when singulaity pesents. We have Lemma 3.. Let u C Dρ\{} be a adially symmetic function that solves the Willmoe suface equation. on Dρ\{} in the xy-plane cented at the oigin. Then f = λ fo some λ R whee f is defined in.9, i.e. F = x, y, u is Willmoe if and only if w w λ + = ϕω + v5, 3. whee 5w ϕω = + w w + w3 3 + w.

9 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION 9 We now establish a uniqueness esult fo an initial value poblem of 3.. Lemma 3.. Thee exists ɛ > such that the equation 3. has at most one solution on, ɛ] with the initial data lim w =, w lim λ log = b Poof. Let w and w be solutions to 3. on, ] fo some < < with 3. satisfied. Let φ = w w. It follows that lim φ = lim φ = + + and φ can be extended to a function in C [, ] with φ = and φ =. Hence lim φ + =, and by integation φ + φ = ϕw t ϕw t + λ v5 t vt 5 dt. t Then φ = ϕw t ϕw t + λ + w 5 + w 5 dt. t Fom 3., we may find Λ = Λ, b, λ such that and Moeove, w i w + w < Λ log. 3.3 w i Λ log, i =,. + w 5 + w 5 C w w C φ log. Applying.4, simila to the poof of.6 by using 3.3 instead of.5, we have ϕw ϕw + λ + w 5 + w 5 C φ + φ log. Then using the aguments in.7, we have φ C whee, ]. Given,, such that φt + tφ t log dt 3.4 t 4C log log + tφ C [,] 4C log log +.

10 J. CHEN & Y. LI If φ is not identically on [, ], then we can find, ], such that Thus, 3.4 yields < φ = φ C [, ]. < φ φ, which is impossible. Theefoe, φ = on [, ], which implies that w = w on [, ɛ] fo ɛ = min{, }. Lemma 3.3. Let w C, a ] be a solution to 3. on, a ]. If lim + w =, then w λ log C [, a ], and consequently lim + w λ log exists. Poof. We assume w < ɛ < on, a], a a, a <. Fist of all, we pove w + w w < C t + w log + w t dt +,, a, 3.5 whee C is a positive constant depends only on a, w a, λ and the uppe bound of w in [, a]. Fo simplicity, we will use C to denote some unifom constant in the est of the poof. Multiplying both sides of 3. by w and then integating fom to a, since wϕ, we have w w λ a w w + w t t v5 w C t dt. Then integating by pats implies w w w w t w C w + t dt, then applying integation by pats again and absobing the tems evaluated at a into C, we have w w w + w a w t w w C + t dt. Reaanging tems, we aive at w + w t dt C w + t dt w w + w. 3.6 To estimate w, integating 3. again, w 5ww = + w + 3w3 + w 5 + λ v5 dt+λ log t t a +w a+ wa a. 3.7 Then, absobing the tems evaluated at a into a unifom constant C, we have w ɛ w + w + C log t. 3.8

11 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION Putting 3.8 into 3.6 and ecalling w < ɛ, we get w + w w a C + t t dt + ɛ w + w t and in tun we conclude the poof of 3.5. Secondly, we pove + C w log + w w < C log. 3.9 By 3.5 and 3.8, w + w C w + log. Fix an ɛ, and choose a such that w < ɛ /C on, a. It is well-known that w w. Theefoe Then Thus w ɛ w + C log,, a. w + ɛ w C log. ɛ w C ɛ log, then ɛ w C t ɛ log t dt C ɛ log ɛ + ɛ and 3.9 is established by dividing ɛ. Now we can finish the poof. Putting 3.9 into 3.5, we ae led to w + w dt < +, t then 5w + w w + 3w3 + w 5 + λ v5 t t dt < Fo convenience, set 5w ψ = + w w + 3w3 + w 5 + λ v5 dt, t t which is continuous in [, a] by 3.. By 3.7, whee w = B = tψtdt + λ log + B, λ log a + w a + wa λ a 4.

12 J. CHEN & Y. LI Thus, lim w λ + log = B + ψ. It is easy to check that the function w λ log is diffeentiable at with deivative B + ψ. Theefoe, w λ log C [, a ]. 3.. Inveted catenoids. A catenoid C c is defined by F c t, θ = c cosh t cos θ, c cosh t sin θ, ct, whee c. The invesion of C c unde the map I a : R 3 R 3 defined in the intoduction is given by c cosh t cos θ c cosh t sin θ x, y, z =,, ct a, R R R whee R = c cosh t + ct a. Let c cosh t ct a ht =, ht =. R R Then I a C c is the suface of evolution geneated by otating the cuve σt = ht,, ht about the z-axis. Since h+ = h =, h has at least citical point. Diect calculation shows that h has only finitely many in fact, at most thee citical points. A tangent vecto of σ is given by σ t = h t,, h t, so t is a citical point of h if and only if the tangent line of σt is paallel to the z-axis. Let t < < t m be the citical points of h and set t =, t m+ = +, then I a F c t i, t i+ S is a gaph ove Dht i+ \Dht i, which cannot be extended as a smooth gaph ove a lage domain, whee i =,, m. Lemma 3.4. Given λ, b R, thee exists a unique pai of numbes c and a, such that Σ c,a is the gaph of a function U with U λ log = b. lim + Moeove, U satisfies the equation 3.. Poof. Afte changing paametization, a catenoid can be witten as ρ cos θ, ρ sin θ, uρ, whee uρ = c accosh ρ c. In this paametiation, R = ρ + uρ a. Suppose Σ c,a is the gaph of U. We have = ρ uρ a, and U = = R R ρ uρ a,, R c,a.

13 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION 3 We now calculate the expansion of U at =. It is clea ρ = uρ a + ρ and By the definition of Σ c,a, uρ c we ae led to U = uρ a + uρ a ρ = uρ a + ρ Ou3 ρ, e uρ c as ρ +. + as ρ +, then by taking log of uρ + e c = cosh uρ ρ = ρ c, uρ = c log ρ + c log c log c c log + e uρ c = c log ρ + c log c log c + o, as ρ +. Using 3., we get uρ a uρ = c log + c log c log c log + + o By 3., = c log + c log c log c + o, as ρ +. U = c log + c log c log c a + o, as. It follows that U + c log is in C [, ρ and Set lim U + c log = c log c log c a 3c. + c = λ 4, a = λ λ log λ 8 b. Then the uniquely detemined Σ c,a is the gaph of U. ρ Poof of Theoem. and Coollay.3. Note that a smooth adially symmetic function f defined on Dρ\{} can be extended at as a function in C Dρ if and only if lim + f =. Poof of Theoem.. Since u C Dρ, lim + w =. By Lemma 3.3, we can wite w λ log = β

14 4 J. CHEN & Y. LI fo some function β C [, ρ. So lim w λ log = β. By Lemma 3.4, thee exist c and a, such that Σ c,a is the gaph of a function U with lim U λ log = β and U satisfies 3.. Then by Lemma 3., U = w in [, ɛ] fo some ɛ. Note that the coefficients in 3. ae smooth on [ɛ, ρ. The standad uniqueness theoem in ODE theoy assets U = w on [ɛ, ρ and consequently U = w on [, ρ. Poof of Coollay.3. By a diect calculation, the fist and the second fundamental foms of F ae + w w g =, A = v w v espectively. Then A w w w w = + = v v v v 3 v and Thus, Dρ\Dɛ A H dµ = Dρ = 4π = 4π = 4π Dρ\Dɛ ρ ɛ ρ ww v 3 d ɛ v vρ vɛ w w v 3 v dµ. A < We now pove lim + w =. Assume thee is a sequence k, such that w k c, whee c may be. Define u k = u k k u k k ρ, w k = u k, F k = x, y, u k, Σ k = F k D \{}. k k Obviously, w k = w k c as k. We may assume that the unit nomal vecto n k p of Σ k at p =,, conveges to a unit vecto α which is not othogonal to the xy-plane, i.e. α,,. 3.5 This is because the angle between the tangent plane T p Σ k and the xy-plane is unifomly bounded away fom when c. Thus, we may assume that Σ k is the gaph of some

15 RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION 5 function ũ k y, z nea p ove the yz-plane. Fo convenience, we set F k = ũ k y, z, y, z and e = F k y. Since Σ k is adially symmetic, the unit cicle {cos θ, sin θ, : θ [, π} is in each Σ k. Then ũ k y, = y and ep =. Fo the second fundamental fom A k of Σ k we have A k ep, ep = Fk y p n kp =,, n k p. 3.6 Howeve, fo any δ >, we have by 3.4 lim k + Dδ A k dµ k = lim k + D k δ Applying the cuvatue estimates in Theoem. in [5], A dµ =. A k p C A k L B 3 p C A k L D, whee Bp 3 = {q R 3 : q p < }. By 3.6, we get,, α =. This contadicts 3.5. Theefoe, we have lim + w =, in tun, u C Dρ. Now the desied esults follow fom Theoem.. Refeences [] Au, T.; Wan, T.: Analysis on an ODE aisen fom studying the shape of a ed blood cell, J. Math. Anal. Appl. 8 3, no., [] Byant, R.: A duality theoem fo Willmoe sufaces. J. Diffeential Geom., [3] Chen, J.; Lamm, T.: A Benstein type theoem fo entie Willmoe gaphs. J. Geom. Anal. 3 3, no., [4] Deckelnick, K.; Dziuk, G.: Eo analysis of a finite element method fo the Willmoe flow of gaphs. Intefaces Fee Bound. 8, [5] Kuwet, E.; Schätzle, R.: The Willmoe flow with small initial enegy. J. Diffeential Geom. 57, no. 3, [6] Luo, Y.; Sun, J.: Remaks on a Benstein type theoem fo entie Willmoe gaphs in R 3. J. Geom. Anal. 4 4, no. 3, [7] Rigoli, M.: On a confomal Benstein type popety. Rend. Sem. Mat. Univ. Politec. Toino , no. 3, Depatment of Mathematics, The Univesity of Bitish Columbia, Vancouve, BC V6TZ, Canada addess: jychen@math.ubc.ca Depatment of Mathematical Sciences, Tsinghua Univesity, Beijing 84, China addess: yxli@math.tsinghua.edu.cn

YUXIANG LI DEPARTMENT OF MATHEMATICAL SCIENCES, TSINGHUA UNIVERSITY, BEIJING , P.R.CHINA.

SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R 3 YUXIANG LI DEPARTMENT OF MATHEMATICAL SCIENCES, TSINGHUA UNIVERSITY, BEIJING 100084, P.R.CHINA. EMAIL: YXLI@MATH.TSINGHUA.EDU.CN. Abstact. Let f : C R