If the soprano wishes to maintain or increase vocal intensity on an /a/ vowel in the range from about E5 and up, all-the-while keeping a vocal quality that closely matches her quality below these notes, she must gradually open her mouth as she ascends above E5. Acoustically she is tuning the first formant of her vocal tract to the fundamental and as the fundamental rises above E5 she must raise the first formant resonance to match the fundamental which she does by opening the mouth. Most female singers will do this instinctively. Middle voice notes below E5 will tune the second formant to one of the overtones of the fundamental.
Here is Anna Netrebko using this technique. Notice how carefully she adjusts mouth opening and lip formation to accommodate correct formant tuning even though she is ostensibly singing an /a/ vowel throughout the piece.
But the above description does not preclude another method of singing these notes above E5. The soprano can continue to leave the mouth opening similar to its opening in the middle voice and attempt to keep the second formant tuned to one of the overtones of the higher notes. This will produce a very fluty sound quality and is often taught by choral directors to keep the soprano section from overpowering the rest of the choir. It is also used in classical singing for particular emotional effects etc. The problem with this kind of formant tuning is that as the fundamental gets higher and higher, the overtones get further and further apart until the only way to tune the second formant to extremely high notes is to round the lips or pucker them a bit and a new quality emerges that resembles a whistle. You can observe some sopranos on youtube doing this when they sing extremely high notes; they actually close the mouth a bit.
Here is Lily Pons using this technique of tuning the second formant to an upper overtone for selected high notes. Notice how she often closes the mouth slightly on these selected high tones exactly when one would expect to see her open the mouth wider. And notice the how sweet and flute like is the resultant tone. Of course, she also uses the technique of tuning the first formant to the fundamental by opening the mouth wider as she goes up the scale when she want a more dramatic quality. By the way, Lily Pons is 76 years old in this performance.
The /a/ vowel is the only viable vowel to sing in the range above G5. It is the vowel that has the highest first formant and the lowest second formant so format tuning of either method is made more possible. On extremely high notes it is sometimes beneficial to attempt a vowel that naturally has a high second formant such as an /e/ or even an /i/ vowel but doing so with the understanding that these vowels will not sound like good /e/ or /i/ vowels because only their second formant is being sounded; their first formant is well below the sung pitch and cannot resonate anything. In other words the tongue is in the correct position for either of these vowels but what is heard is something different from what is expected. Yet when the tongue is in the position for these two vowels it is fronted and the small space in front of the tongue produces the high second formant.