Is the hair that grows beneath the eye part of the beard?

Firstly: 

The saheeh hadeeths indicate that it is obligatory to let the
beard grow and it is haraam to shave it or cut it. 

The scholars of the Standing Committee for Issuing Fatwas
were asked: What is the ruling on shaving the beard or trimming some of it? 

They replied: 

Shaving the beard is haraam, because of the clear saheeh
hadeeths and reports that have been narrated concerning that, and because of
the general meaning of the texts which forbid imitating the disbelievers.
For example, in the hadeeth of Ibn ‘Umar it says that the Messenger of Allah
(blessings and peace of Allah be upon him) said: “Be different from the
mushrikeen: let your beard grow and trim the moustache.” According to
another report: “Trim the moustache and let the beard grow.” And there are
many other similar hadeeths. 

Letting the beard grow means leaving it as it is and leaving
it alone, without shaving or plucking or cutting anything from it. Ibn Hazm
narrated that there was consensus that cutting the moustache and letting the
beard grow is obligatory, and he quoted as evidence a number of hadeeths
including the hadeeth of Ibn ‘Umar (may Allah be pleased with him) quoted
above, as well as the hadeeth of Zayd ibn Arqam, according to which the
Prophet (blessings and peace of Allah be upon him) said: “Whoever does not
trim anything from his moustache is not one of us.” Classed as saheeh by
al-Tirmidhi. In al-Furoo‘ it says: This wording, “according to our
companions” — i.e., the Hanbalis — implies that it is haraam. Shaykh
al-Islam Ibn Taymiyah (may Allah have mercy on him) said: The Qur’aan,
Sunnah and scholarly consensus indicate that it is enjoined to be different
from the disbelievers and it is forbidden to resemble them in general terms,
because resembling them on the outside leads to resembling them in
blameworthy attitudes and actions, and even in beliefs, as it generates love
and friendship on the inside; by the same token, love on the inside
generates resemblance of the outside. Al-Tirmidhi narrated that the
Messenger of Allah (blessings and peace of Allah be upon him) said: “He is
not one of us who resembles people other than us. Do not resemble the Jews
or the Christians …” According to another version: “Whoever imitates a
people is one of them.” this was narrated by Ahmad. And ‘Umar ibn al-Khattab
rejected the testimony of one who plucked his beard.  

Imam Ibn ‘Abd al-Barr said in al-Tamheed:

It is haraam to shave the beard, and no one does that except
effeminate men (mukhannathoon), i.e., those who imitate women. The Prophet
(blessings and peace of Allah be upon him) had a lot of hair in his beard,
as was narrated by Muslim from Jaabir. According to one report, he had a
thick beard and there is another report which is worded differently but the
meaning is the same. It is not permissible to remove any of it because of
the general meaning of the evidence which shows that doing so is forbidden.

End quote from Fataawa al-Lajnah al-Daa’imah, 5/133. 

Secondly: 

The beard is the hair on the cheeks and chin. 

Ibn Manzoor said, quoting from Ibn Sayyidihi: Lahyah (beard)
is a word that includes the hair that grows on the cheeks and chin. 

End quote from Lisaan al-‘Arab, 15/243. 

And it says in al-Qamoos al-Muheet: Lahyah (beard) is
the hair of the cheeks and chin. End quote. 

So what grows beneath the eye is not part of the beard. 

And Allah knows best.