Embedded Delaunay tori and their Willmore energy
EEmbedded Delaunay tori and their Willmore energy
Christian Scharrer ∗ February 23, 2021
Abstract
A family of embedded rotationally symmetric tori in the Euclidean 3-spaceconsisting of two opposite signed constant mean curvature surfaces that converge asvarifolds to a double round sphere is constructed. Using complete elliptic integrals,it is shown that their Willmore energy lies strictly below 8 π . Combining such astrict inequality with previous works by Keller–Mondino–Rivière and
Mondino–Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothlyembedded torus minimising the Willmore functional under isoperimetric constraint,thus completing the solution of the isoperimetric-constrained Willmore problem fortori. Moreover, we deduce the existence of smoothly embedded tori minimising theHelfrich functional with small spontaneous curvature. Furthermore, because of theirsymmetry, the Delaunay tori can be used to construct spheres of high isoperimetricratio, leading to an alternative proof of the known result for the genus zero case.
Given an immersed surface f : Σ → R , the Willmore functional W at f is defined by W ( f ) = Z Σ H d µ, where the mean curvature H is given by the arithmetic mean of the two principalcurvatures, and µ is the Radon measure on Σ corresponding to the pull back metric ofthe Euclidean metric in R along f . The isoperimetric ratio is defined byiso( f ) = area( f )vol( f ) , where area( f ) = Z Σ µ, vol( f ) = 13 Z Σ n · f d µ are the area and enclosed volume, and n : Σ → S is the Gauß map. ∗ University of Warwick. Mathematics Institute. Coventry (UK). Email: [email protected] a r X i v : . [ m a t h . DG ] F e b he aim of this paper is to construct a family of embedded C , regular tori T D ,c in R corresponding to 1 < c < c for some constant c >
1. The tori are rotationallysymmetric and converge for c → Delaunay surfaces : The inner part of the tori hasconstant, strictly positive mean curvature; the outer part has constant, strictly negativemean curvature. These two pieces of Delaunay surfaces have matching normal vectorsalong the curve of intersection, leading to C , regularity of the patched surface. For apicture of the profile curve, see Figure 5. The tori will be called Delaunay tori . Theirmain property is stated in the following theorem which will be proven in Section 6, usingcomplete elliptic integrals.
There exists δ > such that the family of embedded Delaunay tori T D ,c corresponding to < c < δ satisfies W ( T D ,c ) < π whenever < c < δ and lim c → iso( T D ,c ) = ∞ . Denote with S the space of smoothly immersed tori in R . As a consequence of theEuclidean isoperimetric inequality, the isoperimetric ratio is minimised exactly by anyparametrisation of a round sphere. On the other hand, each smoothly embedded closedsurface in R can be smoothly transformed arbitrarily close to a round sphere. Thiscan be done using a one parameter family of Möbius transformations whose centres ofinversion approach a point on the surface. It follows thatiso[ S ] = ( √ π, ∞ )where iso( S ) = √ π . By the regularity result of [KMR14], Theorem 1.1 implies thefollowing 8 π bound for the minimal isoperimetric constrained Willmore energy, see also[MS20b, Remark 1.7]. Let σ > √ π . Then, there holds β ( σ ) := inf {W ( f ) : f ∈ S , iso( f ) = σ } < π. Notice that Corollary 1.2 is stated for all σ ∈ iso[ S ] while Theorem 1.1 only holdsfor high isoperimetric ratios σ . In fact, the crucial part of Corollary 1.2 is exactly that itholds true for high isoperimetric ratios. Indeed, each smoothly immersed closed surfacein R can be smoothly transformed arbitrarily close to the round sphere using onlyconformal transformations, i.e. transformations that do not change the Willmore energy.In this way, one can always scale down the isoperimetric ratio without changing theWillmore energy. This is explained with more details in [MS20b], see also [YC20] for theelementary proof.The following main application will be proven at the end of Section 6; the proof willfollow by combining Theorem 1.1 with previous works of Keller–Mondino–Rivière [KMR14] and
Mondino–Scharrer [MS20b].2 .3 Corollary.
Let σ > √ π . Then, β ( σ ) := inf {W ( f ) : f ∈ S , iso( f ) = σ } is attained by a smoothly embedded minimiser f ∈ S . This completes the solution for the existence (and regularity) problem of isoperimetricconstrained minimisers for the Willmore functional in the genus one case. The genus 0case was solved by
Schygulla [Sch12]. For both cases, minimisers are expected to beunique up to homothety.Given an immersed surface f : Σ → R and c ∈ R , the Helfrich functional H c at f is defined by H c ( f ) = Z Σ ( H − c ) d µ. In order to study the shape of lipid bilayer cell membranes,
Helfrich [Hel73] proposedthe minimisation of H c in the class of closed surfaces with given fixed area and givenfixed volume. The constant c is referred to as spontaneous curvature . Existence ofminimisers in the class of (possibly branched and bubbled) spheres was proven by Mondino–Scharrer [MS20a]. Existence and regularity for minimisers with highergenus remains an open problem. Partial results were obtained by
Choksi–Veneroni [CV13],
Eichmann [Eic20], and
Brazda–Lussardi–Stefanelli [BLS20]. Denote with S the space of smoothly immersed spheres in R and for all σ > √ π let β ( σ ) = inf {W ( f ) : f ∈ S , iso( f ) = σ } . Suppose A , V > A > πV . Then, by Corol-lary 1.2 and [MS20b, Corollary 1.5], the following constant is strictly positive: ε ( A , V ) := q min { π, π + β ( A /V / ) − π } − q β ( A /V / )2 √ A > . (1.1)As another application of Theorem 1.1, the following result on the existence of Helfrichtori will be proven in Section 8. Suppose A , V > satisfy the isoperimetric inequality: A > πV andlet ε = ε ( A , V ) be defined as in Equation (1.1) . Then, for each c ∈ ( − ε, ε ) , there existsa smoothly embedded torus f ∈ S with area( f ) = A , vol( f ) = V and H c ( f ) = inf {H c ( f ) : f ∈ S , area( f ) = A , vol( f ) = V } . In Section 7, it will be shown that the family of Delaunay tori T D ,c can be used toconstruct a family of embedded spheres S D ,c in R with the following property.3 .5 Theorem. Let c > be such that the Delaunay tori T D ,c exist for all < c < c .Then, the family of spheres S D ,c satisfies W ( S D ,c ) = 4 π + W ( T D ,c )2 for all < c < c as well as lim c → iso( S D ,c ) = ∞ . Together with Theorem 1.1, this provides an alternative proof of the known result forspheres [Sch12, Lemma 1].In many classical problems related to the Willmore energy, 8 π bounds such asCorollary 1.2 play a crucial role. One of the reasons is that by the Li-Yau inequality [LY82], any immersed surface with Willmore energy strictly below 8 π is actually embedded.Exemplary for the importance of 8 π bounds is the Willmore flow. Kuwert–Schätzle [KS04] showed that the Willmore flow of spheres exists for all time and converges to around sphere provided the initial surface has Willmore energy less than 8 π . Later, Blatt [Bla09] showed that this energy threshold is actually sharp. Recently,
Dall’Acqua–Müller–Schätzle–Spener [DMSS20] showed that the same holds true for the Willmoreflow of rotationally symmetric tori. Equally important, 8 π bounds are needed in the directmethod of the calculus of variations for the minimisation of the Willmore functional. Thisrelates to the classical Willmore problem, the conformally constrained Willmore problem,and the isoperimetric constrained Willmore problem. In what follows, we illustrate theimportance of 8 π bounds for the Willmore functional in the literature and point outpotential directions for future research.In the early 60s, Willmore [Wil65] showed that the energy now bearing his name,is bounded below by 4 π on the class of closed surfaces, with equality only for the roundsphere. This inequality is sometimes referred to as Willmore’s inequality . More thantwo decades later, in an interesting work that connects Willmore surfaces with minimalsurfaces,
Kusner [Kus89] estimated the area of the celebrated
Lawson surfaces [Law70].
Kusner’s work led to the 8 π bound for the unconstrained minimal Willmore energyamongst surfaces of arbitrary genus. This was one of the key steps in proving existence andregularity of minimisers for the classical minimisation problem proposed by Willmore [Wil65]. Roughly speaking, the 8 π bound prevents macroscopic bubbling in the directmethod of calculus of variations, as by Willmore’s inequality, each of the bubbles wouldcost at least 4 π energy. Indeed, already Simon [Sim93] used the 8 π bound to obtaincompactness (up to suitable Möbius renormalisations) in the so called ambient approach .Later, the 8 π -bound was also used in the parametric approach to obtain compactness inthe moduli space of higher genus surfaces by independent papers of Kuwert–Li [KL12](building on top of previous work of
Müller–Šverák [MŠ95]) and
Rivière [Riv13](building on top of
Hélein’s moving frames technique [Hél02]).The minimisation of the Willmore functional under fixed conformal class was studiedfor instance by
Kuwert–Schätzle [KS13],
Ndiaye–Schätzle [NS14], and
Rivière [Riv14], [Riv15]. Their existence results hold true provided the minimal Willmore energyamongst conformally constrained surfaces lies strictly below 8 π . Using the Willmore flow4or tori of revolution, [DMSS20] showed that each rectangular conformal class contains atorus of revolution with Willmore energy below 8 π .Existence of smoothly embedded isoperimetric constrained Willmore spheres wasproven by Schygulla [Sch12]. Inspired by the computations of
Castro-Villarreal–Guven [CVG07], he applied a family of sphere inversions to a complete catenoid resultingin a family of closed surfaces with arbitrarily high isoperimetric ratios having one point ofmultiplicity two and Willmore energy exactly 8 π . Subsequently he applied the Willmoreflow for a short time around the point of multiplicity two, to obtain a family of surfaceswith Willmore energy strictly below 8 π and arbitrarily high isoperimetric ratios. Thisproved the strict 8 π bound for isoperimetric constrained spheres. Moreover, Schygulla showed that, as varifolds, isoperimetric constrained Willmore spheres (as well as theinverted catenoids) converge to a round sphere of multiplicity 2 as the isoperimetricratio tends to infinity. Notice that the same holds true for the family of Delaunay toriconstructed in this paper. His blow up result was analysed in more detail by
Kuwert–Li [KL18]. They showed that any sequence of isoperimetric constrained Willmore sphereswhose isoperimetric ratios diverge to infinity, indeed converges (up to subsequences,scaling, and translating) to two concentric round spheres of almost the same radiiconnected by a catenoidal neck. These kind of surfaces (i.e. two concentric roundspheres of nearly the same radii connected by one or more catenoidal necks) were alsoconstructed by means of cutting and paste techniques for different purposes in the worksof
Kühnel–Pinkall [KP86],
Müller–Röger [MR14], and
Wojtowytsch [Woj17].One of the advantages of this technique is that it produces not only tori but any surface.However, all of the three examples above have Willmore energy strictly larger than 8 π .It is very natural to expect that, provided higher genus isoperimetric constrainedminimiser exist, the blow up result of Kuwert–Li [KL18] can be generalised to thehigher genus cases. To be more precise, it is expected that any sequence of genus g isoperimetric constrained Willmore surfaces whose isoperimetric ratios diverge to infinity,converges (up to subsequences, scaling, and translating) to two concentric round spheresof nearly the same radii connected by g + 1 catenoidal necks. It is an interesting questionwhether or not the catenoidal necks in the limit have to be distributed over the doublesphere in a certain way. In view of the solution presented in this paper, it is of course verytempting to conjecture that the catenoidal necks have to satisfy a balancing conditionanalogous to the one for constant mean curvature surfaces, see for instance Kapouleas [Kap91], [Kap95], or
Korevaar–Kusner–Solomon [KKS89]. That would mean thatfor tori, the two catenoidal necks necessarily end up being antipodal.
Acknowledgements.
The author is supported by the EPSRC as part of the MAS-DOC DTC at the University of Warwick, grant No. EP/HO23364/1. Moreover, theauthor would like thank
Andrea Mondino and
Filip Rindler for hints and discussionson the subject. 5
Elliptic integrals
Figure 1: Complete elliptic integrals of the first and the second kind.Elliptic integrals are functions defined as the value of common types of integrals thatcannot be expressed in terms of simple functions. They arise when computing geometricquantities such as the arc length of an ellipse or a hyperbola. In particular, they naturallyoccur in the context of constant mean curvature (Delaunay) surfaces of revolution. Thisis because the rotating curves of Delaunay surfaces are given by the roulette generatedby ellipses and hyperbolas. In fact, all the quantities that are needed to construct thefamily of embedded Delaunay tori (see Section 6) as well as their Willmore energy canbe expressed in terms of complete elliptic integrals . Given a so-called elliptic modulus k ,that is a real number 0 < k <
1, the complete elliptic integral of the first kind K and the complete elliptic integral of the second kind E are defined by K ( k ) = Z π/ d θ q − k sin ( θ ) , E ( k ) = Z π/ q − k sin ( θ ) d θ. All the formulas for elliptic integrals used in this paper can be found in the book of
Byrd–Friedman [BF71]. The derivatives are given byd K ( k )d k = E ( k ) k (1 − k ) − K ( k ) k , d E ( k )d k = E ( k ) − K ( k ) k . The
Gauß transformation works as follows. Define the complementary modulus k andthe transformed modulus k by k = p − k , k = 1 − k k . Then, there holds (see [BF71, 164.02]) K ( k ) = (1 + k ) K ( k ) , E ( k ) = (1 + k ) E ( k ) − k (1 + k ) K ( k ) . (2.1)6oreover, K grows like log(1 /k ), namelylim k → − (cid:16) K ( k ) − log(4 / p − k ) (cid:17) = 0 (2.2)and E is bounded: 1 ≤ E ≤ π/ . (2.3) A surface of revolution in R is given by a parametrisation X of the type X ( t, θ ) = ( f ( t ) cos( θ ) , f ( t ) sin( θ ) , g ( t ))with parameters t lying in an open interval and 0 ≤ θ ≤ π , where f, g are real valuedfunctions. The rotating curve c := ( f, g ) is referred to as meridian or profile curve . Theunderlying geometry is described by the coefficients of the first fundamental form E = X t · X t = ˙ f + ˙ g = | ˙ c | , F = X t · X θ = 0 , G = X θ · X θ = f and the second fundamental form L = X tt · n = ˙ f ¨ g − ¨ f ˙ g | ˙ c | , M = X tθ · n = 0 , N = X θθ · n = f ˙ g | ˙ c | where the Gauß map n is given by n = X t × X θ | X t × X θ | . The mean curvature H is defined as the arithmetic mean of the principal curvatures κ , κ , that is 2 H = κ + κ = LE + NG = ˙ f ¨ g − ¨ f ˙ g | ˙ c | + ˙ gf | ˙ c | . In this paper, we will focus on surfaces of revolution with constant mean curvature H = 12 a (3.1)for some given 0 = a ∈ R . These surfaces arise as critical points of the volume constrainedarea functional. Outside of a discrete set, one has ˙ g = 0 and thus c ( ϕ ( t )) = ( ρ ( t ) , t )for some parameter transform ϕ and some real valued function ρ . Hence, outside of adiscrete set, Equation (3.1) can be turned into a second order ODE. Its solutions werefirst described by Delaunay [Del41] and are now named after him. More precisely,solutions for a > unduloids and will be discussed in Section 4; solutions for a < nodoids and will be discussed in Section 5.7
Unduloids
Figure 2: Profile curve of an unduloid with 2 periods, a = 1, b = 0 . f, g ) is given by the roulette of an ellipse with generating pointbeing one of the foci. To be more precise, let a > b > c = √ a − b . Then,the equation x a + y b = 1describes a standard ellipse centred at the origin with width 2 a , height 2 b , and foci ± c on the x -axis. Rolling the ellipse without slipping along a line, each of the two focuspoints will describe a periodic curve, the roulette . Let ( f, g ) be the parametrisation ofone of the roulettes such that the period is 2 π . Bendito–Bowick–Medina [BBM14]found the following representation: f ( t ) = b a − c cos( t ) p a − c cos ( t ) (4.1) g ( t ) = Z t q a − c cos ( x ) d x − c sin( t ) a − c cos( t ) p a − c cos ( t ) (4.2)with coefficients of the first fundamental form E = a b ( a + c cos( t )) , G = b a − c cos( t ) a + c cos( t ) , and mean curvature H = 12 a . (4.3)The extrema are given bymin image f = f (0) = a − c, max image f = f ( π ) = a + c. (4.4)8ext, a formula for the area will be determined. Using the Weierstraß substitution,the area A of the rotational symmetric surface corresponding to one period can becomputed by Z π Z π √ EG d θ d t = 2 πab Z π s a − c cos( t )( a + c cos( t )) d t = 4 πab Z π s a + c cos( t )( a − c cos( t )) d t = 4 πab Z ∞ vuut a + c − x x ( a − c − x x ) x x = 8 πab Z ∞ vuut a (1 + x ) + c (1 − x ) (cid:0) a (1 + x ) − c (1 − x ) (cid:1) d x = 8 πab Z ∞ vuut ( a + c ) + ( a − c ) x (cid:0) ( a − c ) + ( a + c ) x (cid:1) d x = 8 πab s a − c ( a + c ) Z ∞ s ˜ a + t (˜ b + t ) d t for ˜ a = ( a + c ) / ( a − c ) and ˜ b = ( a − c ) / ( a + c ). The last integral can be transformed intoa complete elliptic integral of the second kind using [BF71, 221.01] with k = 1 − ˜ b / ˜ a and g = 1 / ˜ a : Z ∞ s ˜ a + t (˜ b + t ) d t = gk E ( k ) . One can show that k = 4 ac ( a + c ) , k = a − ca + c , gk = ( a + c ) / ( a − c ) / and thus A = 8 πa ( a + c ) E ( k ); k = 2 √ aca + c . (4.5)Notice that this coincides with the area formula for unduloids computed for a differentparametrisation in [HMO07] and [MO03].Finally, we compute the extrinsic length L of one period. It is given by L = | g (2 π ) − g (0) | = a Z π s − c a cos ( x ) d x = 4 a Z π/ s(cid:16) − c a (cid:17) + c a sin ( x ) d x = 4 a s − c a Z π/ q n sin ( x ) d x for n = c/b . Letting k = n / (1 + n ), (282.03) and (315.02) in [BF71] imply Z π/ q n sin ( x ) d x = 1 k E ( k ) . Thus, since k = ca , k = s − c a , it follows that L = 4 aE ( k ); k = ca . (4.6)9 Nodoids
Figure 3: Separate roulettes ( f ± , g ± )(bottom/top) for a = b = 1. Figure 4: Both roulettes patched to-gether for 2 periods and a = b = 1.Nodoids are surfaces of revolution with constant, strictly negative mean curvature. Theirrotating curve ( f, g ) is given by the roulette of a hyperbola with generating points givenby the foci. To be more precise, let a, b > c = √ a + b . Then, the equation x a − y b = 1describes a hyperbola in canonical form with distance a to the centre and foci ± c onthe x -axis. Rolling the right branch of the hyperbola without slipping along a line, eachof the two focus points will describe a curve, the roulette . Bendito–Bowick–Medina [BBM14] found parametrisations ( f ± , g ± ) of the roulettes, where ( f + , g + ) corresponds tothe focus ( c,
0) and ( f − , g − ) is the reflected roulette corresponding to the focus ( − c, f ± ( t ) = b c cosh( t ) ∓ a q c cosh ( t ) − a (5.1) g ± ( t ) = Z t q c cosh ( x ) − a d x − c sinh( t ) c cosh( t ) ∓ a q c cosh ( t ) − a (5.2)with parameter t running through all of R , coefficients of the first fundamental form E ± = a b ( c cosh( t ) ± a ) , G ± = b c cosh( t ) ∓ ac cosh( t ) ± a , and mean curvature H ± = − a . (5.3)One has max image f + = lim t →±∞ f + ( t ) = b, min image f + = f + (0) = c − a min image f − = lim t →±∞ f − ( t ) = b max image f − = f − (0) = c + a (5.4)10nd thus, after translation along the axis of rotation, the two roulettes corresponding tothe foci ( ± c,
0) can be glued together into one periodic curve (see Figure 4).Next, the area A of the rotational symmetric surface corresponding to one periodwill be computed. Using the parameter transformations t = artanh( x ) and x = sin( t ),one infers Z ∞−∞ p E + G + d t = ab Z ∞−∞ s c cosh( t ) − a ( c cosh( t ) + a ) d t = 2 ab Z vuuut c √ − x − a ( c √ − x + a ) d x − x = 2 ab Z vuut c − a √ − x ( c + a √ − x ) d x √ − x = 2 ab Z π/ s c − a cos( t )( c + a cos( t )) d t = 2 ab Z ππ/ s c + a cos( t )( c − a cos( t )) d t and similarly, Z ∞−∞ p E − G − d t = 2 ab Z π/ s c + a cos( t )( c − a cos( t )) d t. Consequently, A = Z π Z ∞−∞ p E + G + + p E − G − d t d θ = 4 πab Z π s c + a cos( t )( c − a cos( t )) d t. In Section 4 it was shown that Z π s c + a cos( t )( c − a cos( t )) d t = 2 E ( k ) c − a ; k = 2 √ aca + c . Thus, it follows that A = 8 πa ( a + c ) E ( k ); k = 2 √ aca + c . (5.5)Next, in order to determine the extrinsic length L of one period, a parametertransformation will be carried out. First, for α = a /c , u = tan t , and s = arsinh( u ),one computes f ± ( s ) = b c √ u ∓ a p c (1 + u ) − a = b c ∓ a cos( t ) p c − a cos ( t ) , (5.6)for − π/ < t < π/
2, and Z s q cosh ( x ) − α d x = Z u √ x − α d x √ x = Z u r − α x d x = Z t p − α cos ( x ) d x cos ( x ) = − Z t α cos( x ) sin( x ) p − α cos ( x ) tan( x ) d x + q − α cos ( t ) tan( t )= −√ α Z t sin ( x ) d x q sin ( x ) + b /a + tan( t ) q − α cos ( t ) ,
11s well astan( t ) s − a c cos ( t ) − c ∓ a cos( t ) p c − a cos ( t ) = ac sin( t ) ± c − a cos( t ) p c − a cos ( t ) . It follows g ± ( s ) = − a Z t sin ( x ) d x q sin ( x ) + b /a ± a sin( t ) s c ∓ a cos( t ) c ± a cos( t ) (5.7)for − π/ < t < π/
2. Notice that, up to translation along the axis of rotation, bothcurves R → R with s ( f + ( s ) , g + ( s )) and s ( f − ( s ) , g − ( s )) as given by (5.6), (5.7)parametrise the whole periodic curve resulting from the patched roulettes in Figure 4.Finally, we compute the extrinsic length L of one period. It is given by L = | g + ( s ( π/ − g + ( s ( − π/ | + | g − ( s ( π/ − g − ( s ( − π/ | = 4 an Z π/ sin ( x ) d x q n sin ( x )for n = a /b . Letting k = n / (1 + n ), it follows k = ac , k = bc , ank k k = c. Therefore, by (282.04) and (318.02) in [BF71], L = 4 c h E ( k ) − k K ( k ) i ; k = ac . (5.8) In this Section, we will construct the family of embedded Delaunay tori T D ,c with1 < c < c for some constant c and we will prove Theorem 1.1. Moreover, at the end ofthis section, we will give a proof of Corollary 1.3.Each Delaunay torus is a rotationally symmetric surfaces whose profile curve (seeFigure 5) consists of one period of an unduloid roulette (see Figure 2) and one period ofa nodoid roulette (see Figure 4). The construction works as follows. Start with a oneparameter family of patched nodoids (see Section 5) running for one period and startingat the minimum according to (5.4), where a = 1 and c > b is given by b = √ c − c −
1. Next, dependingon the parameter c , find a > y > ± y , width 2 a , and height 2 b for b = p a − y (see Section 4) running for oneperiod and starting at its minimum a − y according (4.4), fits right into the given nodoid.That means the two end points where the patched nodoids reach their minimum need tomatch the two end points where the unduloid reaches its minimum. Notice that, in thisway, the profile curve is C , regular. The coordinates of the two patching points can12 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.250.500.751.001.251.501.752.00 Figure 5: Profile curve Delaunay toruswith c = 1 . Figure 6: Energy curve for the family ofDelaunay tori and 8 π bound.be determined using the equations (4.4), (4.6) for the unduloid and (5.4), (5.8) for thenodoid. Thus, a, y are given as the solution of the system of equations aE (cid:16) ya (cid:17) = 4 c (cid:20) E (cid:16) c (cid:17) − (cid:16) − c (cid:17) K (cid:16) c (cid:17)(cid:21) a − y = c − . (6.1)(6.2)Abbreviating ε := c − , L := c (cid:20) E (cid:16) c (cid:17) − ε c + 1 c K (cid:16) c (cid:17)(cid:21) , the system of equation reads as ( y + ε ) E (cid:16) yy + ε (cid:17) = La = y + ε. Define the function F : (1 , ∞ ) × (0 , ∞ ) → R , F ( c, y ) = ( y + ε ( c )) E (cid:16) yy + ε ( c ) (cid:17) − L ( c ) . There holds ε log 4 p − /c = ε log 4 c √ ε √ c + 1 ≤ c √ c + 1 √ ε c ↓ −−→ . Hence, by (2.2), lim c → L ( c ) = 1 . (6.3)Moreover, ∂ c L = [ E ( c ) − (1 − c ) K ( c )] + c [ E ( c ) − K ( c )] c − c + c h − c K ( c ) − (1 − c )[ E ( c ) / (1 − c ) − K ( c )] c ( − c ) i = E ( c ) − K ( c )13nd, for k = y/ ( y + ε ) ∂ c ( F + L ) = E ( k ) + ( y + ε )[ E ( k ) − K ( k )] y + εy − y ( y + ε ) = K ( k )which implies ∂ c F = K (cid:16) yy + ε (cid:17) + K (cid:16) c (cid:17) − E (cid:16) c (cid:17) . Writing k = y/ ( y + ε ), there holds ∂ y F = E ( k ) + ( y + ε )[ E ( k ) − K ( k )] y + εy ε ( y + ε ) = (1 + εy ) E ( k ) − εy K ( k ) . Moreover, for fixed y , ε log 4 p − y / ( y + ε ) = ε log 4( y + ε ) p εy + ε ≤ y + ε ) √ y + ε √ ε c ↓ −−→ c → ∂ y F ( c, y ) = 1 . Hence, using (6.3) and (2.3), it follows that there exists c > < c < c there exists a unique y ( c ) > F ( c, y ( c )) = 0 and derivative y = E ( c ) − K ( c ) − K ( yy + ε )( y + ε ) E ( yy + ε ) − εK ( yy + ε ) y. (6.5)Using (4.3), (4.5) for the unduloid and (5.3), (5.5) for the nodoid, one obtains theWillmore energy of the Delaunay tori T D ,c : W ( T D ,c ) = W nod + W und (6.6)where W nod = 2 π (1 + c ) E (cid:16) √ c c (cid:17) , W und = 2 π (cid:16) yy + ε (cid:17) E (cid:16) p y ( y + ε )2 y + ε (cid:17) . Using (2.3) and (6.3), one can see that C ≤ y ≤ /C for some C > c → yy + ε = 1 , lim c → y = 1 , lim c → εK (cid:16) yy + ε (cid:17) = 0 , lim c → W ( T D ,c ) = 8 π. Next, we show that ∂ c W ( T D ,c ) < c close to 1 which then implies W ( T D ,c ) < π for c close to 1. First, we compute ∂ c W nod . For this purpose, let k = 2 √ c/ (1 + c ). Then, ∂ c k = 1(1 + c ) √ c − √ c (1 + c ) = 1 − c (1 + c ) √ c ∂ c W nod = 2 πE ( k ) + 2 π (1 + c )[ E ( k ) − K ( k )] 1 + c √ c − c (1 + c ) √ c = π (cid:16)(cid:16) c (cid:17) E ( k ) + (cid:16) − c (cid:17) K ( k ) (cid:17) . By the Gauß transformation (2.1) there holds k = s − c (1 + c ) = c − c + 1 , k = 1 − k k = 1 c , and ∂ c W nod π = (1 + c ) h (1 + c − c +1 ) E ( c ) − c − c +1 (1 + c ) K ( c ) i + (1 − c )(1 + c ) K ( c ) = 2 E ( c ) . Hence, ∂ c W nod = 2 πE (cid:16) c (cid:17) . (6.7)In order to compute ∂ y W und , let k = 2 p y ( y + ε ) / (2 y + ε ). Then, there holds ∂ y k = y + ε (2 y + ε ) √ y ( y + ε ) − √ y ( y + ε )(2 y + ε ) = ε (2 y + ε ) √ y ( y + ε ) , ∂ y yy + ε = ε ( y + ε ) and ∂ y W und = 2 π ε ( y + ε ) E ( k ) + 2 π y + εy + ε [ E ( k ) − K ( k )] y + ε √ y ( y + ε ) ε √ y ( y + ε )(2 y + ε ) = πε ( y + ε ) (cid:0) E ( k ) + εy [ E ( k ) − K ( k )] (cid:1) = πεy ( y + ε ) (cid:0) (2 y + ε ) E ( k ) − εK ( k ) (cid:1) . By the Gauß transformation there holds k = s − y ( y + ε )(2 y + ε ) = ε y + ε , k = 1 − k k = yy + ε and(2 y + ε ) E ( k ) − εK ( k ) = (2 y + ε )[(1 + ε y + ε ) E ( yy + ε ) − ε y + ε (1 + yy + ε ) K ( yy + ε )] − ε (1 + yy + ε ) K ( yy + ε ) = 2 (cid:0) ( y + ε ) E ( yy + ε ) − ε (1 + yy + ε ) K ( yy + ε ) (cid:1) . (6.8)Therefore, ∂ y W und = 2 πεy ( y + ε ) (cid:20) ( y + ε ) E (cid:16) yy + ε (cid:17) − ε (cid:16) yy + ε (cid:17) K (cid:16) yy + ε (cid:17)(cid:21) . (6.9)Abbreviate z = y /y , and k = y/ ( y + ε ). Then, (6.9) and (6.5) imply ∂ y W und · y = πεy ( y + ε ) h ( y + ε ) E ( k ) − ε (1 + yy + ε ) K ( k ) i · E ( c ) − K ( c ) − K ( k )( y + ε ) E ( k ) − εK ( k ) y = πε ( y + ε ) h E ( c ) − K ( c ) − K ( yy + ε ) i − πε ( y + ε ) εyy + ε zK ( yy + ε ) . (6.10)15inally we compute ∂ ε W und . For this purpose let k = 2 p y ( y + ε ) / (2 y + ε ). Then,there holds ∂ ε k = y (2 y + ε ) p y ( y + ε ) − p y ( y + ε )(2 y + ε ) = − yε (2 y + ε ) p y ( y + ε )and ∂ ε W und = 2 π − y ( y + ε ) E ( k ) + 2 π y + εy + ε [ E ( k ) − K ( k )] y + ε √ y ( y + ε ) − yε (2 y + ε ) √ y ( y + ε ) = − π ( y + ε ) (cid:0) (2 y + ε ) E ( yy + ε ) − εK ( yy + ε ) (cid:1) . Thus, by (6.8), ∂ ε W und = − π ( y + ε ) (cid:20) ( y + ε ) E (cid:16) yy + ε (cid:17) − ε (cid:16) yy + ε (cid:17) K (cid:16) yy + ε (cid:17)(cid:21) . Recall that, by the choice of y , there holds( y + ε ) E (cid:16) yy + ε (cid:17) = c (cid:20) E (cid:16) c (cid:17) − (cid:16) − c (cid:17) K (cid:16) c (cid:17)(cid:21) . Therefore, ∂ ε W und = − π ( y + ε ) (cid:20) cE (cid:16) c (cid:17) − ε c + 1 c K (cid:16) c (cid:17) − ε (cid:16) yy + ε (cid:17) K (cid:16) yy + ε (cid:17)(cid:21) . (6.11)Putting (6.7), (6.10), and (6.11) into (6.6), it follows ∂ c W ( T D ,c ) = ∂ c W nod + ∂ ε W und + ∂ y W und · y = 2 πE ( c ) − π ( y + ε ) h cE (cid:16) c (cid:17) − ε c +1 c K (cid:16) c (cid:17) − ε (cid:16) yy + ε (cid:17) K (cid:16) yy + ε (cid:17)i + πε ( y + ε ) h E ( c ) − K ( c ) − K ( yy + ε ) i − πε y ( y + ε ) zK ( yy + ε )= 2 πE ( c ) h − c ( y + ε ) + ε ( y + ε ) i + πε ( y + ε ) K ( c ) h c +1 c − i + πεy ( y + ε ) K ( yy + ε )[1 − εz ]= 2 π (1 − y + ε ) ) E ( c ) + πc ( y + ε ) εK ( c ) + πy ( y + ε ) (1 − εz ) εK ( yy + ε ) . Therefore, ∂ c W ( T D ,c )2 π = a (cid:16) − y + ε ) (cid:17) + a εK (cid:16) c (cid:17) + a εK (cid:16) yy + ε (cid:17) (6.12)where a , a , a → c →
1. In particular, ∂ c W ( T D ,c ) → c →
1. We claim thatlim c → εK ( c ) + εK ( yy + ε )1 − y + ε ) = − . (6.13)First, recall that εK ( c ) c ↓ −−→ , εK ( yy + ε ) c ↓ −−→ , εy c ↓ −−→ , y c ↓ −−→ −∞ , y c ↓ −−→ .
16t follows ε y ∂ c K ( c ) = ε y (cid:20) E ( c )1 − c − K ( c ) (cid:21) c − c c ↓ −−→ ,∂ c yy + ε = y ∂ y yy + ε + ∂ ε yy + ε = εy − y ( y + ε ) c ↓ −−→ − , ε y ∂ c K ( yy + ε ) = ε y " E ( yy + ε )1 − y y + ε )2 − K ( yy + ε ) y + εy ∂ c yy + ε c ↓ −−→ . Thus, by L’Hôspital’s rule,lim c → εK ( c ) + εK ( yy + ε )1 − y + ε ) = lim c → K ( c ) + K ( yy + ε ) y + ε ) [1 + y ] = − ∂ c W ( T D ,c ) < c close to 1 . Therefore, for some δ >
0, there holds W ( T D ,c ) < π whenever 1 < c < δ. (6.14)The Delaunay tori T D ,c converge to a round sphere of multiplicity 2 and radius 2 as c → c → iso( T D ,c ) = ∞ . Together with (6.14), this proves Theorem 1.1. It remains to mention that, since the twoperiodic profile curves of nodoids and unduloids are patched together at their minimum,the resulting Delaunay torus is a C , regular closed genus-1 surface. Proof of Corollary 1.2.
Let E S be the space of Lipschitz immersions of S into R as defined in Section 2.2 of [KMR14]. Similarly, let T be an abstract 2-dimensionaltorus and denote with E T the space of Lipschitz immersions of T into R (see [KMR14,Section 2.2]). By Schygulla [Sch12] and [KMR14, Theorem 1.1] the following holdstrue. For each σ > √ π , there exists a smoothly embedded spherical surface S S ,σ with W ( S S ,σ ) = β ( σ ) := inf {W ( ~ Φ) : ~ Φ ∈ E S , iso( ~ Φ) = σ } . Moreover, by Theorem 1.6 in
Keller–Mondino–Rivière [KMR14], for each σ in theset I := σ ∈ R : inf ~ Φ ∈E T iso( ~ Φ)= σ W ( ~ Φ) < min { π, π + β ( σ ) − π } ⊂ ( √ π, ∞ )17here exists a smoothly embedded torus Σ in R with W (Σ ) = β ( σ ) := inf ~ Φ ∈E T iso( ~ Φ)= σ W ( ~ Φ) . From Corollary 1.5 in
Mondino–Scharrer [MS20b] it follows I = { σ ∈ R : β ( σ ) < π } . Recall that the function β ( · ) is non-decreasing on the set I . This can be shown usingMöbius transformations, see for instance [YC20, Theorem 3.1]. Therefore, the set I is infact an interval. Moreover, each C , embedding of T into R is in particular a Lipschitzimmersion, i.e. a member of the space E T . Thus, by Theorem 1.1, I = ( √ π, ∞ ) . Figure 7: Profile curve of half a Delau-nay torus with c = 1 . Figure 8: Concentric quarter circles fit-ting into half a Delaunay torus.The Delaunay tori T D ,c corresponding to 1 < c < c can be used to constructspheres with analogous properties. The first part of the construction works just like theconstruction of the Delaunay tori only that now, both the nodoid and the unduloid runonly for half a period instead of one full period. To be more precise, both the nodoidand the unduloid now only run from their minimum according to (4.4), (5.4) until theyreach their maximum (according to (4.4), (5.4)) but not until they reach their minimumagain. This results in half a Delaunay torus, see Figure 7. Notice that unduloids and18odoids are symmetric around their maxima ( t = π in (4.1), (4.2) for unduloids; t = 0 in(5.1), (5.2) for nodoids). Thus, the Willmore energy of this particular half of a Delaunaytorus is indeed half the Willmore energy of a whole Delaunay torus. Let c, y, a be thebalancing parameters according to (6.1) and (6.2). Then the maxima of the nodoid andthe unduloid are given by c + 1 and a + y , respectively. Next, take two concentric circularsectors with radii c + 1 and a + y both of which being one quarter of a full circle, seeFigure 8. Choose the centre of the two circular sectors at L/ L = 4 aE ( y/a ) (see (4.6)). Then, the two circular sectorsfit right into the half Delaunay torus, resulting in a C , curve. Since the two circularsectors meet the axis of rotation perpendicular, the resulting surface of revolution is C , regular too. It is of sphere type. The full profile can be seen in Figure 9. The resultingfamily of surfaces is called Delaunay spheres . Since a + y, c + 1 c ↓ −−→
2, the Delaunayspheres converge as varifolds to a sphere of multiplicity 2 as c →
1. Their Willmoreenergy is given by W ( S D ,c ) = 4 π + W ( T D ,c )2 . Figure 9: Profile of a Delaunay sphere with c = 1 . Suppose A , V > A > πV and let S ( A , V ) = { f ∈ S : area( f ) = A , vol( f ) = V } . c = 0, then the Helfrich functional reduces to the Willmore functional: H = W . Moreover, one can show (see [MS20a, Equation 4.26]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) inf f ∈S ( A ,V ) q W ( f ) − inf f ∈S ( A ,V ) q H c ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | c | p A . (8.1)In particular, the minimal Helfrich energy is continuous with respect to c at c = 0. Forthe case c = 0, existence of smoothly embedded minimisers with given fixed area andvolume corresponds to Corollary 1.3 in [KMR14]. Theorem 1.4 states that minimisersremain embedded for c close to zero. Moreover, (by the choice of ε ( A , V ) in (1.1)) aminimising sequence for c = 0 has the same uniform bounds on the Willmore energy asa minimising sequence for c close to zero. Indeed, we will see that the compactness proofin [KMR14] still works for c close to zero. However, for general c , minimisers are nolonger embedded, see [MS20a]. The following proof is a combination of four independentresults: the two strict inequalities Theorem 1.1 and [MS20b, Corollary 1.5] are needed todeduce that ε ( A , V ) (as defined in (1.1)) is strictly positive; then, one can apply thecompactness proof of [KMR14]; finally, one can conclude the regularity from [MS20a](after Rivière [Riv08]).In order to prove Theorem 1.4, let T be an abstract 2-dimensional torus and let E T be the space of Lipschitz immersions of T into R as defined in [KMR14, Section 2.2].Let ~ Φ k be a minimising sequence ofinf {H c ( ~ Φ) : ~ Φ ∈ E T , area( ~ Φ) = A , vol( ~ Φ) = V } . Recall the definition of ε ( A , V ) in Equation (1.1): ε ( A , V ) := q min { π, π + β ( A /V / ) − π } − q β ( A /V / )2 √ A . By Theorem 1.1 and [MS20b, Corollary 1.5], there holds ε ( A , V ) >
0. Using thecontinuity property (8.1) one can show for | c | < ε ( A , V ) that (see the proof of Lemma4.4 in [MS20a]) lim sup k →∞ q W ( ~ Φ k ) < q β ( A /V / ) + 2 ε ( A , V ) p A . Thus, by the definition of ε ( A , V ) it followslim sup k →∞ W ( ~ Φ k ) < π (8.2)and lim sup k →∞ W ( ~ Φ k ) < π + β ( A /V / ) − π. (8.3)In Section 4.3 of [KMR14] it is shown that due to the strict inequality in (8.3), theconformal factors of ~ Φ k are bounded away from finitely many concentration points20 , . . . , a N in T . Hence, by the uniform energy bound in (8.2), there exists ~ Φ ∞ ∈ E T such that (after passing to a subsequence and after re-parametrising) for all δ > ~ Φ k → ~ Φ ∞ as k → ∞ weakly in W , ( T \ S Ni =1 B δ ( a i ) , R ). (8.4)Moreover, in Section 4.2 of [KMR14] it is shown that due to (8.2), there holds thatlim k →∞ vol( ~ Φ k ) = vol( ~ Φ ∞ ) , lim k →∞ area( ~ Φ k ) = area( ~ Φ ∞ ) . After the mentioned re-parametrisations, the ~ Φ k ’s are weakly conformal which implies∆ k ~ Φ k = 2 H ~ Φ k for the intrinsic Laplacian ∆ k . Therefore, by the weak convergence (8.4)it follows that for all δ > k →∞ Z T \ S Ni =1 B δ ( a i ) H ~ Φ k d µ ~ Φ k = Z T \ S Ni =1 B δ ( a i ) H ~ Φ ∞ d µ ~ Φ ∞ . Moreover, by [KMR14, Equation (4.7)],lim inf δ → lim inf k →∞ Z B δ ( a i ) µ ~ Φ k = 0for all i ∈ { , . . . , N } . Using the Cauchy–Schwarz inequality and the uniform bound onthe Willmore energy (8.2), it follows that after passing to a subsequencelim k →∞ Z T H ~ Φ k d µ ~ Φ k = Z T H ~ Φ ∞ d µ ~ Φ ∞ . Thus, by lower semi continuity of the Willmore functional under the convergence of (8.4), H c ( ~ Φ ∞ ) ≤ lim inf k →∞ H c ( ~ Φ k ) , W ( ~ Φ ∞ ) < π. Therefore, ~ Φ ∞ is a minimiser and, by the Li–Yau inequality, ~ Φ ∞ ∈ W , ( T , R ) is an em-bedding without branch points. Moreover, by the regularity result [MS20a, Theorem 4.3](after [Riv08]), ~ Φ ∞ ∈ C ∞ ( T , R ) which completes the proof of Theorem 1.4. References [BBM14] Enrique Bendito, Mark J. Bowick, and Agustín Medina. A natural parameterization of theroulettes of the conics generating the Delaunay surfaces.
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