ON RELATED VECTOR FIELDS OF CAPILLARY SURFACES

In this paper, some necessary and sufficient conditions are given for the related vector fields of capillary surfaces to be Killing, conformal Killing, and homothetic conformal Killing vectors in the n-dimensional domain Ω, and a construction of capillary surfaces is also given by means of the related vector fields.

A.K. OZOK a surface S: u U(Xl,...,x ), of constant mean curvature H, defined over a n region R in n-dimensional Euclidean space n > 2,  It is shown by R. Finn [2] that, in any situation for which the problem [(i.i), (1.2)] has a solution, there exists a vector field (x) Vu (cos )/ + IVu (1.3) in , satisfying l div o in R (1.4) v .i on l (1.5) The symbols , Z are used here to denote alternatively both a set and its measure.
So we have a Euclidean vector field in an n-dimensional domain , n > 2.
We consider as a part of n-dimensional Euclidean space En, with the funda- mental positive definite metric n ds 2 d i,J=l lj dxidxj covered by any system of coordinate neighborhoods (xi).It is well known that some special kind of vectors (for example, Killing, conformal Killing, homothetic conformal Killing vectors) define motions (isometries) in a metric space [3], [4], [5], [6], [7].The purpose of this paper is to determine for which kind of "capillary surfaces" the related vector field (1.3) in defines the above men- tioned motions.A_d in Section 4, a construction of capillary surfaces is given by means of related vector fields.
2. THE CONDITIONS FOR THE VECTOR FIELD (1.3) TO BE A KILLING VECTOR IN .
The vector field (1.3) can be written in the form u+/-(x) Jl(X) (i=l,...,n) ( (cos )/i + u + -. +u n where u I u/x I. On the other hand, the condition for the vector field l(X) to be a Killing vector is (2.4) cos y THEOREM i.If the problem [(I.i), (1.2)] has a solution H const. 0, then the related vector field (1.3) in fl cannot be a Killing vector.Thus the vector field i(x) cannot generate a local one-parameter group of motions in ft.ll) If the problem [(I.i), (1.2)] has a solution H 0, i.e., if we have a minimal capillary surface, by virtue of equation (2.1),we find from (2.2) the following conditions . (UikU juk+u where u u u Uk x-and Uki Uik XkXi xixk On the other hand, for the surface S: u U(Xl,...,Xn), the coefficients of the first and second fundamental forms are defined So equations (2.5) and (2.6) take the forms respectively: (2.9) The above equations (2.8) and (2.9) can also be put in the form 2)] has a solution H 0, then the necessary and sufficient condition for the related vector field (1.3) to be a Killing vector in the n-dimensional domain where gij and dij are the coefficients of the first and second fundamental forms of the minimal capillary surface u(x).
THEOREM 3.For the minimal capillary surfaces for which the coefficients of the first and second fundamental forms satisfy the condition (2.10), the related vector field (1.3) generates a local one-parameter group of motions in the n-di- mensional domain APPLICATION i.In the case n 2, we find from equation (2.10) and g22dll g12d12 glld22 g12d12 (2.11) (gll+g22)d12 (dll+d22)g12 (2.12)So we have A.K. OZOK _I d d_A/ P.
(2.13) Hence, the minimal capillary surface must be a totally umbilical surface.But the sphere is the only surface, all points of which are umbillcs.So, if n 2, the minimal capillary surface does not exist such that the related vector field (1.3) is a Killing vector in .Then from this result and Theorem i we obtain THEOREM 4. In the case n 2, in any situation for which the problem (i.i), (1. 2)] has a solution, the related vector field (1.3) in cannot be a Killing vector; thus the vector field (x) cannot generate a local one-parameter group of motions in .
3. THE CONDITIONS FOR THE VECTOR FIELD (1..3) TO BE A CONFORMAL KILLING AND HOMO- THETIC CONFORMAL KILLING VECTOR IN .
The condition for the vector field i(x) to be a conformal Killing vector is where is a certain scalar function.Since equation (3.1) gives Cp --1 div ob substituting equation (2.4) into (3.2),we find H constant.
By virtue of equation (2.1), we find from equation (3.1)

.s)
Thus we have THEOREM 5.The necessary and sufficient condition for the vector field (1.3) to be a homothetic conformal Killing vector in the n-dimensional domain where gij and dij are the coefficients of the first and second fundamental forms of the capillary surface u(x).Thus we have THEOREM 7. In the case n 2, the only capillary surface for which the related vector field (1.3) is a homothetic conformal Killing vector in is a sphere.THEOREM 8.In the case n 2, if the solution of the problem [(i,i), ( is a sphere, the related vector field (1.3) generates a local one-parameter group of homothetic motions in .
Integrating equation (3.1), we can also find the form of a homothetlc con- where r, cos y and m i are constants.So, i(x) has the form (3.9).
Thus we have THEOREM 9.For an arbitrary n, if the solution of the problem [(I.i), (1.2)] is a sphere, the related vector field (1.3) is a homothetlc conformal Killing vec-  tor in the n-dimensional domain R, n > 2.
And we have also THEOREM I0.Fr an arbitrary n, if the solution of the problem [(i.i), (1.2)] is a sphere, the related vector field (1.3) generates a local one-parameter group of homothetic motions in the n-dimensional domain , n > 2.
A.K. OZOK 4. CONSTRUCTION OF CAPILLARY SURFACES BY MEANS OF RELATED VECTOR FIELDS.
i) If we have a homothetic conformal Killing vector of the form i(x) exi +i (i =l,...,n; 's are const.)where v is the exterior unit normal on Z, so we may construct a capillary sur- face S: u u(x), defined over the base domain , such that S meets vertical cylinder walls z over I in a constant angle prescribed by cos y < I and then it has the constant mean curvature H a cos y.Indeed, it is easily seen that this vector field (4.1) satisfies the conditions (1.4), (1.5)  ii) In general, if a vector field o3i(x) satisfies the following two con- ditions with the constant div o3, 3o3.
i .___3.s2y I (.__3_x k o3io3k 1 Ok mjmk)-0 (i,j =l,...,n) then we can construct a capillary surface S, defined over the base domain such that S meets vertical cylinder walls z over Z B in a constant angle i prescribed by cos y < luIo31 and then it has the constant mean curvature , cos y div o3.n Indeed, since equation (4.7) is the condition of integrability of equation (4.3) and the given vector field o3i(x) also satisfies the conditions (1.4), (1.5)   and (1.6), the related capillary surface S: u u(x) can be found from equation (4.3).
ACKNOWLEDGMENT.This work was completed while the author was a visiting scholar at Stanford University.

3 .
Let the lower hemisphere, centered at the point M(ml,...,mn+I) wlth radius r, be a solution of the problem [(i.i),(1) -" [r cos y xi r cos y] (i--1,...,n) such that S meets vertical cylinder walls z over E DR in a (prescribed) constant angle Y.One is led on 7..Here v is the exterior unit normal on Y.. (For physical and geometrical background information, see [i].) For the capillary surfaces for which the coefficients of the first and second fundamental forms satisfy the condition (3.8), the related and (1.6), so we may consider this field related to the capillary